# uncompyle6 version 3.2.3
# Python bytecode 3.6 (3379)
# Decompiled from: Python 3.6.8 |Anaconda custom (64-bit)| (default, Feb 21 2019, 18:30:04) [MSC v.1916 64 bit (AMD64)]
# Embedded file name: _pydecimal.py
"""
This is an implementation of decimal floating point arithmetic based on
the General Decimal Arithmetic Specification:

    http://speleotrove.com/decimal/decarith.html

and IEEE standard 854-1987:

    http://en.wikipedia.org/wiki/IEEE_854-1987

Decimal floating point has finite precision with arbitrarily large bounds.

The purpose of this module is to support arithmetic using familiar
"schoolhouse" rules and to avoid some of the tricky representation
issues associated with binary floating point.  The package is especially
useful for financial applications or for contexts where users have
expectations that are at odds with binary floating point (for instance,
in binary floating point, 1.00 % 0.1 gives 0.09999999999999995 instead
of 0.0; Decimal('1.00') % Decimal('0.1') returns the expected
Decimal('0.00')).

Here are some examples of using the decimal module:

>>> from decimal import *
>>> setcontext(ExtendedContext)
>>> Decimal(0)
Decimal('0')
>>> Decimal('1')
Decimal('1')
>>> Decimal('-.0123')
Decimal('-0.0123')
>>> Decimal(123456)
Decimal('123456')
>>> Decimal('123.45e12345678')
Decimal('1.2345E+12345680')
>>> Decimal('1.33') + Decimal('1.27')
Decimal('2.60')
>>> Decimal('12.34') + Decimal('3.87') - Decimal('18.41')
Decimal('-2.20')
>>> dig = Decimal(1)
>>> print(dig / Decimal(3))
0.333333333
>>> getcontext().prec = 18
>>> print(dig / Decimal(3))
0.333333333333333333
>>> print(dig.sqrt())
1
>>> print(Decimal(3).sqrt())
1.73205080756887729
>>> print(Decimal(3) ** 123)
4.85192780976896427E+58
>>> inf = Decimal(1) / Decimal(0)
>>> print(inf)
Infinity
>>> neginf = Decimal(-1) / Decimal(0)
>>> print(neginf)
-Infinity
>>> print(neginf + inf)
NaN
>>> print(neginf * inf)
-Infinity
>>> print(dig / 0)
Infinity
>>> getcontext().traps[DivisionByZero] = 1
>>> print(dig / 0)
Traceback (most recent call last):
  ...
  ...
  ...
decimal.DivisionByZero: x / 0
>>> c = Context()
>>> c.traps[InvalidOperation] = 0
>>> print(c.flags[InvalidOperation])
0
>>> c.divide(Decimal(0), Decimal(0))
Decimal('NaN')
>>> c.traps[InvalidOperation] = 1
>>> print(c.flags[InvalidOperation])
1
>>> c.flags[InvalidOperation] = 0
>>> print(c.flags[InvalidOperation])
0
>>> print(c.divide(Decimal(0), Decimal(0)))
Traceback (most recent call last):
  ...
  ...
  ...
decimal.InvalidOperation: 0 / 0
>>> print(c.flags[InvalidOperation])
1
>>> c.flags[InvalidOperation] = 0
>>> c.traps[InvalidOperation] = 0
>>> print(c.divide(Decimal(0), Decimal(0)))
NaN
>>> print(c.flags[InvalidOperation])
1
>>>
"""
__all__ = [
    "Decimal",
    "Context",
    "DecimalTuple",
    "DefaultContext",
    "BasicContext",
    "ExtendedContext",
    "DecimalException",
    "Clamped",
    "InvalidOperation",
    "DivisionByZero",
    "Inexact",
    "Rounded",
    "Subnormal",
    "Overflow",
    "Underflow",
    "FloatOperation",
    "DivisionImpossible",
    "InvalidContext",
    "ConversionSyntax",
    "DivisionUndefined",
    "ROUND_DOWN",
    "ROUND_HALF_UP",
    "ROUND_HALF_EVEN",
    "ROUND_CEILING",
    "ROUND_FLOOR",
    "ROUND_UP",
    "ROUND_HALF_DOWN",
    "ROUND_05UP",
    "setcontext",
    "getcontext",
    "localcontext",
    "MAX_PREC",
    "MAX_EMAX",
    "MIN_EMIN",
    "MIN_ETINY",
    "HAVE_THREADS",
]
__xname__ = __name__
__name__ = "decimal"
__version__ = "1.70"
__libmpdec_version__ = "2.4.2"
import math as _math, numbers as _numbers, sys

try:
    from collections import namedtuple as _namedtuple

    DecimalTuple = _namedtuple("DecimalTuple", "sign digits exponent")
except ImportError:
    DecimalTuple = lambda *args: args

ROUND_DOWN = "ROUND_DOWN"
ROUND_HALF_UP = "ROUND_HALF_UP"
ROUND_HALF_EVEN = "ROUND_HALF_EVEN"
ROUND_CEILING = "ROUND_CEILING"
ROUND_FLOOR = "ROUND_FLOOR"
ROUND_UP = "ROUND_UP"
ROUND_HALF_DOWN = "ROUND_HALF_DOWN"
ROUND_05UP = "ROUND_05UP"
HAVE_THREADS = True
if sys.maxsize == 9223372036854775807:
    MAX_PREC = 999999999999999999
    MAX_EMAX = 999999999999999999
    MIN_EMIN = -999999999999999999
else:
    MAX_PREC = 425000000
    MAX_EMAX = 425000000
    MIN_EMIN = -425000000
MIN_ETINY = MIN_EMIN - (MAX_PREC - 1)


class DecimalException(ArithmeticError):
    """Base exception class.
    
    Used exceptions derive from this.
    If an exception derives from another exception besides this (such as
    Underflow (Inexact, Rounded, Subnormal) that indicates that it is only
    called if the others are present.  This isn't actually used for
    anything, though.
    
    handle  -- Called when context._raise_error is called and the
               trap_enabler is not set.  First argument is self, second is the
               context.  More arguments can be given, those being after
               the explanation in _raise_error (For example,
               context._raise_error(NewError, '(-x)!', self._sign) would
               call NewError().handle(context, self._sign).)
    
    To define a new exception, it should be sufficient to have it derive
    from DecimalException.
    """

    def handle(self, context, *args):
        pass


class Clamped(DecimalException):
    """Exponent of a 0 changed to fit bounds.
    
    This occurs and signals clamped if the exponent of a result has been
    altered in order to fit the constraints of a specific concrete
    representation.  This may occur when the exponent of a zero result would
    be outside the bounds of a representation, or when a large normal
    number would have an encoded exponent that cannot be represented.  In
    this latter case, the exponent is reduced to fit and the corresponding
    number of zero digits are appended to the coefficient ("fold-down").
    """

    pass


class InvalidOperation(DecimalException):
    """An invalid operation was performed.
    
    Various bad things cause this:
    
    Something creates a signaling NaN
    -INF + INF
    0 * (+-)INF
    (+-)INF / (+-)INF
    x % 0
    (+-)INF % x
    x._rescale( non-integer )
    sqrt(-x) , x > 0
    0 ** 0
    x ** (non-integer)
    x ** (+-)INF
    An operand is invalid
    
    The result of the operation after these is a quiet positive NaN,
    except when the cause is a signaling NaN, in which case the result is
    also a quiet NaN, but with the original sign, and an optional
    diagnostic information.
    """

    def handle(self, context, *args):
        if args:
            ans = _dec_from_triple(args[0]._sign, args[0]._int, "n", True)
            return ans._fix_nan(context)
        else:
            return _NaN


class ConversionSyntax(InvalidOperation):
    """Trying to convert badly formed string.
    
    This occurs and signals invalid-operation if a string is being
    converted to a number and it does not conform to the numeric string
    syntax.  The result is [0,qNaN].
    """

    def handle(self, context, *args):
        return _NaN


class DivisionByZero(DecimalException, ZeroDivisionError):
    """Division by 0.
    
    This occurs and signals division-by-zero if division of a finite number
    by zero was attempted (during a divide-integer or divide operation, or a
    power operation with negative right-hand operand), and the dividend was
    not zero.
    
    The result of the operation is [sign,inf], where sign is the exclusive
    or of the signs of the operands for divide, or is 1 for an odd power of
    -0, for power.
    """

    def handle(self, context, sign, *args):
        return _SignedInfinity[sign]


class DivisionImpossible(InvalidOperation):
    """Cannot perform the division adequately.
    
    This occurs and signals invalid-operation if the integer result of a
    divide-integer or remainder operation had too many digits (would be
    longer than precision).  The result is [0,qNaN].
    """

    def handle(self, context, *args):
        return _NaN


class DivisionUndefined(InvalidOperation, ZeroDivisionError):
    """Undefined result of division.
    
    This occurs and signals invalid-operation if division by zero was
    attempted (during a divide-integer, divide, or remainder operation), and
    the dividend is also zero.  The result is [0,qNaN].
    """

    def handle(self, context, *args):
        return _NaN


class Inexact(DecimalException):
    """Had to round, losing information.
    
    This occurs and signals inexact whenever the result of an operation is
    not exact (that is, it needed to be rounded and any discarded digits
    were non-zero), or if an overflow or underflow condition occurs.  The
    result in all cases is unchanged.
    
    The inexact signal may be tested (or trapped) to determine if a given
    operation (or sequence of operations) was inexact.
    """

    pass


class InvalidContext(InvalidOperation):
    """Invalid context.  Unknown rounding, for example.
    
    This occurs and signals invalid-operation if an invalid context was
    detected during an operation.  This can occur if contexts are not checked
    on creation and either the precision exceeds the capability of the
    underlying concrete representation or an unknown or unsupported rounding
    was specified.  These aspects of the context need only be checked when
    the values are required to be used.  The result is [0,qNaN].
    """

    def handle(self, context, *args):
        return _NaN


class Rounded(DecimalException):
    """Number got rounded (not  necessarily changed during rounding).
    
    This occurs and signals rounded whenever the result of an operation is
    rounded (that is, some zero or non-zero digits were discarded from the
    coefficient), or if an overflow or underflow condition occurs.  The
    result in all cases is unchanged.
    
    The rounded signal may be tested (or trapped) to determine if a given
    operation (or sequence of operations) caused a loss of precision.
    """

    pass


class Subnormal(DecimalException):
    """Exponent < Emin before rounding.
    
    This occurs and signals subnormal whenever the result of a conversion or
    operation is subnormal (that is, its adjusted exponent is less than
    Emin, before any rounding).  The result in all cases is unchanged.
    
    The subnormal signal may be tested (or trapped) to determine if a given
    or operation (or sequence of operations) yielded a subnormal result.
    """

    pass


class Overflow(Inexact, Rounded):
    """Numerical overflow.
    
    This occurs and signals overflow if the adjusted exponent of a result
    (from a conversion or from an operation that is not an attempt to divide
    by zero), after rounding, would be greater than the largest value that
    can be handled by the implementation (the value Emax).
    
    The result depends on the rounding mode:
    
    For round-half-up and round-half-even (and for round-half-down and
    round-up, if implemented), the result of the operation is [sign,inf],
    where sign is the sign of the intermediate result.  For round-down, the
    result is the largest finite number that can be represented in the
    current precision, with the sign of the intermediate result.  For
    round-ceiling, the result is the same as for round-down if the sign of
    the intermediate result is 1, or is [0,inf] otherwise.  For round-floor,
    the result is the same as for round-down if the sign of the intermediate
    result is 0, or is [1,inf] otherwise.  In all cases, Inexact and Rounded
    will also be raised.
    """

    def handle(self, context, sign, *args):
        if context.rounding in (
            ROUND_HALF_UP,
            ROUND_HALF_EVEN,
            ROUND_HALF_DOWN,
            ROUND_UP,
        ):
            return _SignedInfinity[sign]
        if sign == 0:
            if context.rounding == ROUND_CEILING:
                return _SignedInfinity[sign]
            return _dec_from_triple(
                sign, "9" * context.prec, context.Emax - context.prec + 1
            )
        if sign == 1:
            if context.rounding == ROUND_FLOOR:
                return _SignedInfinity[sign]
            return _dec_from_triple(
                sign, "9" * context.prec, context.Emax - context.prec + 1
            )


class Underflow(Inexact, Rounded, Subnormal):
    """Numerical underflow with result rounded to 0.
    
    This occurs and signals underflow if a result is inexact and the
    adjusted exponent of the result would be smaller (more negative) than
    the smallest value that can be handled by the implementation (the value
    Emin).  That is, the result is both inexact and subnormal.
    
    The result after an underflow will be a subnormal number rounded, if
    necessary, so that its exponent is not less than Etiny.  This may result
    in 0 with the sign of the intermediate result and an exponent of Etiny.
    
    In all cases, Inexact, Rounded, and Subnormal will also be raised.
    """

    pass


class FloatOperation(DecimalException, TypeError):
    """Enable stricter semantics for mixing floats and Decimals.
    
    If the signal is not trapped (default), mixing floats and Decimals is
    permitted in the Decimal() constructor, context.create_decimal() and
    all comparison operators. Both conversion and comparisons are exact.
    Any occurrence of a mixed operation is silently recorded by setting
    FloatOperation in the context flags.  Explicit conversions with
    Decimal.from_float() or context.create_decimal_from_float() do not
    set the flag.
    
    Otherwise (the signal is trapped), only equality comparisons and explicit
    conversions are silent. All other mixed operations raise FloatOperation.
    """

    pass


_signals = [
    Clamped,
    DivisionByZero,
    Inexact,
    Overflow,
    Rounded,
    Underflow,
    InvalidOperation,
    Subnormal,
    FloatOperation,
]
_condition_map = {
    ConversionSyntax: InvalidOperation,
    DivisionImpossible: InvalidOperation,
    DivisionUndefined: InvalidOperation,
    InvalidContext: InvalidOperation,
}
_rounding_modes = (
    ROUND_DOWN,
    ROUND_HALF_UP,
    ROUND_HALF_EVEN,
    ROUND_CEILING,
    ROUND_FLOOR,
    ROUND_UP,
    ROUND_HALF_DOWN,
    ROUND_05UP,
)
try:
    import threading
except ImportError:

    class MockThreading(object):
        def local(self, sys=sys):
            return sys.modules[__xname__]

    threading = MockThreading()
    del MockThreading

try:
    threading.local
except AttributeError:
    if hasattr(threading.current_thread(), "__decimal_context__"):
        del threading.current_thread().__decimal_context__

    def setcontext(context):
        """Set this thread's context to context."""
        if context in (DefaultContext, BasicContext, ExtendedContext):
            context = context.copy()
            context.clear_flags()
        threading.current_thread().__decimal_context__ = context

    def getcontext():
        """Returns this thread's context.
        
        If this thread does not yet have a context, returns
        a new context and sets this thread's context.
        New contexts are copies of DefaultContext.
        """
        try:
            return threading.current_thread().__decimal_context__
        except AttributeError:
            context = Context()
            threading.current_thread().__decimal_context__ = context
            return context


else:
    local = threading.local()
    if hasattr(local, "__decimal_context__"):
        del local.__decimal_context__

    def getcontext(_local=local):
        """Returns this thread's context.
        
        If this thread does not yet have a context, returns
        a new context and sets this thread's context.
        New contexts are copies of DefaultContext.
        """
        try:
            return _local.__decimal_context__
        except AttributeError:
            context = Context()
            _local.__decimal_context__ = context
            return context

    def setcontext(context, _local=local):
        """Set this thread's context to context."""
        if context in (DefaultContext, BasicContext, ExtendedContext):
            context = context.copy()
            context.clear_flags()
        _local.__decimal_context__ = context

    del threading
    del local


def localcontext(ctx=None):
    """Return a context manager for a copy of the supplied context
    
    Uses a copy of the current context if no context is specified
    The returned context manager creates a local decimal context
    in a with statement:
        def sin(x):
             with localcontext() as ctx:
                 ctx.prec += 2
                 # Rest of sin calculation algorithm
                 # uses a precision 2 greater than normal
             return +s  # Convert result to normal precision
    
         def sin(x):
             with localcontext(ExtendedContext):
                 # Rest of sin calculation algorithm
                 # uses the Extended Context from the
                 # General Decimal Arithmetic Specification
             return +s  # Convert result to normal context
    
    >>> setcontext(DefaultContext)
    >>> print(getcontext().prec)
    28
    >>> with localcontext():
    ...     ctx = getcontext()
    ...     ctx.prec += 2
    ...     print(ctx.prec)
    ...
    30
    >>> with localcontext(ExtendedContext):
    ...     print(getcontext().prec)
    ...
    9
    >>> print(getcontext().prec)
    28
    """
    if ctx is None:
        ctx = getcontext()
    return _ContextManager(ctx)


class Decimal(object):
    """Floating point class for decimal arithmetic."""

    __slots__ = ("_exp", "_int", "_sign", "_is_special")

    def __new__(cls, value="0", context=None):
        r"""Create a decimal point instance.
        
        >>> Decimal('3.14')              # string input
        Decimal('3.14')
        >>> Decimal((0, (3, 1, 4), -2))  # tuple (sign, digit_tuple, exponent)
        Decimal('3.14')
        >>> Decimal(314)                 # int
        Decimal('314')
        >>> Decimal(Decimal(314))        # another decimal instance
        Decimal('314')
        >>> Decimal('  3.14  \n')        # leading and trailing whitespace okay
        Decimal('3.14')
        """
        self = object.__new__(cls)
        if isinstance(value, str):
            m = _parser(value.strip().replace("_", ""))
            if m is None:
                if context is None:
                    context = getcontext()
                return context._raise_error(
                    ConversionSyntax, "Invalid literal for Decimal: %r" % value
                )
            if m.group("sign") == "-":
                self._sign = 1
            else:
                self._sign = 0
            intpart = m.group("int")
            if intpart is not None:
                fracpart = m.group("frac") or ""
                exp = int(m.group("exp") or "0")
                self._int = str(int(intpart + fracpart))
                self._exp = exp - len(fracpart)
                self._is_special = False
            else:
                diag = m.group("diag")
                if diag is not None:
                    self._int = str(int(diag or "0")).lstrip("0")
                    if m.group("signal"):
                        self._exp = "N"
                    else:
                        self._exp = "n"
                else:
                    self._int = "0"
                    self._exp = "F"
                self._is_special = True
            return self
        if isinstance(value, int):
            if value >= 0:
                self._sign = 0
            else:
                self._sign = 1
            self._exp = 0
            self._int = str(abs(value))
            self._is_special = False
            return self
        if isinstance(value, Decimal):
            self._exp = value._exp
            self._sign = value._sign
            self._int = value._int
            self._is_special = value._is_special
            return self
        if isinstance(value, _WorkRep):
            self._sign = value.sign
            self._int = str(value.int)
            self._exp = int(value.exp)
            self._is_special = False
            return self
        if isinstance(value, (list, tuple)):
            if len(value) != 3:
                raise ValueError(
                    "Invalid tuple size in creation of Decimal from list or tuple.  The list or tuple should have exactly three elements."
                )
            if not (isinstance(value[0], int) and value[0] in (0, 1)):
                raise ValueError(
                    "Invalid sign.  The first value in the tuple should be an integer; either 0 for a positive number or 1 for a negative number."
                )
            self._sign = value[0]
            if value[2] == "F":
                self._int = "0"
                self._exp = value[2]
                self._is_special = True
            else:
                digits = []
                for digit in value[1]:
                    if isinstance(digit, int):
                        if 0 <= digit <= 9:
                            if digits or digit != 0:
                                digits.append(digit)
                            raise ValueError(
                                "The second value in the tuple must be composed of integers in the range 0 through 9."
                            )

                if value[2] in ("n", "N"):
                    self._int = ("").join(map(str, digits))
                    self._exp = value[2]
                    self._is_special = True
                else:
                    if isinstance(value[2], int):
                        self._int = ("").join(map(str, digits or [0]))
                        self._exp = value[2]
                        self._is_special = False
                    else:
                        raise ValueError(
                            "The third value in the tuple must be an integer, or one of the strings 'F', 'n', 'N'."
                        )
            return self
        if isinstance(value, float):
            if context is None:
                context = getcontext()
            context._raise_error(
                FloatOperation,
                "strict semantics for mixing floats and Decimals are enabled",
            )
            value = Decimal.from_float(value)
            self._exp = value._exp
            self._sign = value._sign
            self._int = value._int
            self._is_special = value._is_special
            return self
        raise TypeError("Cannot convert %r to Decimal" % value)

    @classmethod
    def from_float(cls, f):
        """Converts a float to a decimal number, exactly.
        
        Note that Decimal.from_float(0.1) is not the same as Decimal('0.1').
        Since 0.1 is not exactly representable in binary floating point, the
        value is stored as the nearest representable value which is
        0x1.999999999999ap-4.  The exact equivalent of the value in decimal
        is 0.1000000000000000055511151231257827021181583404541015625.
        
        >>> Decimal.from_float(0.1)
        Decimal('0.1000000000000000055511151231257827021181583404541015625')
        >>> Decimal.from_float(float('nan'))
        Decimal('NaN')
        >>> Decimal.from_float(float('inf'))
        Decimal('Infinity')
        >>> Decimal.from_float(-float('inf'))
        Decimal('-Infinity')
        >>> Decimal.from_float(-0.0)
        Decimal('-0')
        
        """
        if isinstance(f, int):
            return cls(f)
        if not isinstance(f, float):
            raise TypeError("argument must be int or float.")
        if _math.isinf(f) or _math.isnan(f):
            return cls(repr(f))
        if _math.copysign(1.0, f) == 1.0:
            sign = 0
        else:
            sign = 1
        n, d = abs(f).as_integer_ratio()
        k = d.bit_length() - 1
        result = _dec_from_triple(sign, str(n * 5 ** k), -k)
        if cls is Decimal:
            return result
        else:
            return cls(result)

    def _isnan(self):
        """Returns whether the number is not actually one.
        
        0 if a number
        1 if NaN
        2 if sNaN
        """
        if self._is_special:
            exp = self._exp
            if exp == "n":
                return 1
            if exp == "N":
                return 2
            return 0

    def _isinfinity(self):
        """Returns whether the number is infinite
        
        0 if finite or not a number
        1 if +INF
        -1 if -INF
        """
        if self._exp == "F":
            if self._sign:
                return -1
            return 1
        else:
            return 0

    def _check_nans(self, other=None, context=None):
        """Returns whether the number is not actually one.
        
        if self, other are sNaN, signal
        if self, other are NaN return nan
        return 0
        
        Done before operations.
        """
        self_is_nan = self._isnan()
        if other is None:
            other_is_nan = False
        else:
            other_is_nan = other._isnan()
        if self_is_nan or other_is_nan:
            if context is None:
                context = getcontext()
            if self_is_nan == 2:
                return context._raise_error(InvalidOperation, "sNaN", self)
            if other_is_nan == 2:
                return context._raise_error(InvalidOperation, "sNaN", other)
            if self_is_nan:
                return self._fix_nan(context)
            return other._fix_nan(context)
        else:
            return 0

    def _compare_check_nans(self, other, context):
        """Version of _check_nans used for the signaling comparisons
        compare_signal, __le__, __lt__, __ge__, __gt__.
        
        Signal InvalidOperation if either self or other is a (quiet
        or signaling) NaN.  Signaling NaNs take precedence over quiet
        NaNs.
        
        Return 0 if neither operand is a NaN.
        
        """
        if context is None:
            context = getcontext()
        if self._is_special or other._is_special:
            pass
        if self.is_snan():
            return context._raise_error(
                InvalidOperation, "comparison involving sNaN", self
            )
        elif other.is_snan():
            return context._raise_error(
                InvalidOperation, "comparison involving sNaN", other
            )
        elif self.is_qnan():
            return context._raise_error(
                InvalidOperation, "comparison involving NaN", self
            )
        elif other.is_qnan():
            return context._raise_error(
                InvalidOperation, "comparison involving NaN", other
            )
        else:
            return 0

    def __bool__(self):
        """Return True if self is nonzero; otherwise return False.
        
        NaNs and infinities are considered nonzero.
        """
        return self._is_special or self._int != "0"

    def _cmp(self, other):
        """Compare the two non-NaN decimal instances self and other.
        
        Returns -1 if self < other, 0 if self == other and 1
        if self > other.  This routine is for internal use only."""
        if self._is_special or other._is_special:
            self_inf = self._isinfinity()
            other_inf = other._isinfinity()
            if self_inf == other_inf:
                return 0
            if self_inf < other_inf:
                return -1
            return 1
        if not self:
            if not other:
                return 0
            return -(-1) ** other._sign
        if not other:
            return (-1) ** self._sign
        if other._sign < self._sign:
            return -1
        if self._sign < other._sign:
            return 1
        self_adjusted = self.adjusted()
        other_adjusted = other.adjusted()
        if self_adjusted == other_adjusted:
            self_padded = self._int + "0" * (self._exp - other._exp)
            other_padded = other._int + "0" * (other._exp - self._exp)
            if self_padded == other_padded:
                return 0
            if self_padded < other_padded:
                return -(-1) ** self._sign
            return (-1) ** self._sign
        else:
            if self_adjusted > other_adjusted:
                return (-1) ** self._sign
            return -(-1) ** self._sign

    def __eq__(self, other, context=None):
        self, other = _convert_for_comparison(self, other, equality_op=True)
        if other is NotImplemented:
            return other
        elif self._check_nans(other, context):
            return False
        else:
            return self._cmp(other) == 0

    def __lt__(self, other, context=None):
        self, other = _convert_for_comparison(self, other)
        if other is NotImplemented:
            return other
        else:
            ans = self._compare_check_nans(other, context)
            if ans:
                return False
            return self._cmp(other) < 0

    def __le__(self, other, context=None):
        self, other = _convert_for_comparison(self, other)
        if other is NotImplemented:
            return other
        else:
            ans = self._compare_check_nans(other, context)
            if ans:
                return False
            return self._cmp(other) <= 0

    def __gt__(self, other, context=None):
        self, other = _convert_for_comparison(self, other)
        if other is NotImplemented:
            return other
        else:
            ans = self._compare_check_nans(other, context)
            if ans:
                return False
            return self._cmp(other) > 0

    def __ge__(self, other, context=None):
        self, other = _convert_for_comparison(self, other)
        if other is NotImplemented:
            return other
        else:
            ans = self._compare_check_nans(other, context)
            if ans:
                return False
            return self._cmp(other) >= 0

    def compare(self, other, context=None):
        """Compare self to other.  Return a decimal value:
        
        a or b is a NaN ==> Decimal('NaN')
        a < b           ==> Decimal('-1')
        a == b          ==> Decimal('0')
        a > b           ==> Decimal('1')
        """
        other = _convert_other(other, raiseit=True)
        if self._is_special or other and other._is_special:
            ans = self._check_nans(other, context)
            if ans:
                return ans
            return Decimal(self._cmp(other))

    def __hash__(self):
        """x.__hash__() <==> hash(x)"""
        if self._is_special:
            if self.is_snan():
                raise TypeError("Cannot hash a signaling NaN value.")
            else:
                if self.is_nan():
                    return _PyHASH_NAN
                if self._sign:
                    return -_PyHASH_INF
                return _PyHASH_INF
            if self._exp >= 0:
                exp_hash = pow(10, self._exp, _PyHASH_MODULUS)
            else:
                exp_hash = pow(_PyHASH_10INV, -self._exp, _PyHASH_MODULUS)
            hash_ = int(self._int) * exp_hash % _PyHASH_MODULUS
            ans = hash_ if self >= 0 else -hash_
            if ans == -1:
                return -2
            return ans

    def as_tuple(self):
        """Represents the number as a triple tuple.
        
        To show the internals exactly as they are.
        """
        return DecimalTuple(self._sign, tuple(map(int, self._int)), self._exp)

    def as_integer_ratio(self):
        """Express a finite Decimal instance in the form n / d.
        
        Returns a pair (n, d) of integers.  When called on an infinity
        or NaN, raises OverflowError or ValueError respectively.
        
        >>> Decimal('3.14').as_integer_ratio()
        (157, 50)
        >>> Decimal('-123e5').as_integer_ratio()
        (-12300000, 1)
        >>> Decimal('0.00').as_integer_ratio()
        (0, 1)
        
        """
        if self._is_special:
            if self.is_nan():
                raise ValueError("cannot convert NaN to integer ratio")
            else:
                raise OverflowError("cannot convert Infinity to integer ratio")
            if not self:
                return (0, 1)
            n = int(self._int)
            if self._exp >= 0:
                n, d = n * 10 ** self._exp, 1
            else:
                d5 = -self._exp
                while d5 > 0:
                    if n % 5 == 0:
                        n //= 5
                        d5 -= 1

                d2 = -self._exp
                shift2 = min((n & -n).bit_length() - 1, d2)
                if shift2:
                    n >>= shift2
                    d2 -= shift2
                d = 5 ** d5 << d2
            if self._sign:
                n = -n
            return (n, d)

    def __repr__(self):
        """Represents the number as an instance of Decimal."""
        return "Decimal('%s')" % str(self)

    def __str__(self, eng=False, context=None):
        """Return string representation of the number in scientific notation.
        
        Captures all of the information in the underlying representation.
        """
        sign = ["", "-"][self._sign]
        if self._is_special:
            if self._exp == "F":
                return sign + "Infinity"
            if self._exp == "n":
                return sign + "NaN" + self._int
            return sign + "sNaN" + self._int
        leftdigits = self._exp + len(self._int)
        if self._exp <= 0:
            pass
        if leftdigits > -6:
            dotplace = leftdigits
        if not eng:
            dotplace = 1
        else:
            if self._int == "0":
                dotplace = (leftdigits + 1) % 3 - 1
            else:
                dotplace = (leftdigits - 1) % 3 + 1
            if dotplace <= 0:
                intpart = "0"
                fracpart = "." + "0" * -dotplace + self._int
            else:
                if dotplace >= len(self._int):
                    intpart = self._int + "0" * (dotplace - len(self._int))
                    fracpart = ""
                else:
                    intpart = self._int[:dotplace]
                    fracpart = "." + self._int[dotplace:]
                if leftdigits == dotplace:
                    exp = ""
                else:
                    if context is None:
                        context = getcontext()
                    exp = ["e", "E"][context.capitals] + "%+d" % (leftdigits - dotplace)
                return sign + intpart + fracpart + exp

    def to_eng_string(self, context=None):
        """Convert to a string, using engineering notation if an exponent is needed.
        
        Engineering notation has an exponent which is a multiple of 3.  This
        can leave up to 3 digits to the left of the decimal place and may
        require the addition of either one or two trailing zeros.
        """
        return self.__str__(eng=True, context=context)

    def __neg__(self, context=None):
        """Returns a copy with the sign switched.
        
        Rounds, if it has reason.
        """
        if self._is_special:
            ans = self._check_nans(context=context)
            if ans:
                return ans
            if context is None:
                context = getcontext()
            if not self:
                if context.rounding != ROUND_FLOOR:
                    ans = self.copy_abs()
                ans = self.copy_negate()
            return ans._fix(context)

    def __pos__(self, context=None):
        """Returns a copy, unless it is a sNaN.
        
        Rounds the number (if more than precision digits)
        """
        if self._is_special:
            ans = self._check_nans(context=context)
            if ans:
                return ans
            if context is None:
                context = getcontext()
            if not self:
                if context.rounding != ROUND_FLOOR:
                    ans = self.copy_abs()
                ans = Decimal(self)
            return ans._fix(context)

    def __abs__(self, round=True, context=None):
        """Returns the absolute value of self.
        
        If the keyword argument 'round' is false, do not round.  The
        expression self.__abs__(round=False) is equivalent to
        self.copy_abs().
        """
        if not round:
            return self.copy_abs()
        else:
            if self._is_special:
                ans = self._check_nans(context=context)
                if ans:
                    return ans
                if self._sign:
                    ans = self.__neg__(context=context)
                else:
                    ans = self.__pos__(context=context)
            return ans

    def __add__(self, other, context=None):
        """Returns self + other.
        
        -INF + INF (or the reverse) cause InvalidOperation errors.
        """
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        if context is None:
            context = getcontext()
        if self._is_special or other._is_special:
            ans = self._check_nans(other, context)
            if ans:
                return ans
            if self._isinfinity():
                if self._sign != other._sign:
                    if other._isinfinity():
                        return context._raise_error(InvalidOperation, "-INF + INF")
                    return Decimal(self)
                if other._isinfinity():
                    return Decimal(other)
                exp = min(self._exp, other._exp)
                negativezero = 0
                if context.rounding == ROUND_FLOOR:
                    if self._sign != other._sign:
                        negativezero = 1
                if not self:
                    if not other:
                        sign = min(self._sign, other._sign)
                        if negativezero:
                            sign = 1
                        ans = _dec_from_triple(sign, "0", exp)
                        ans = ans._fix(context)
                        return ans
                    if not self:
                        exp = max(exp, other._exp - context.prec - 1)
                        ans = other._rescale(exp, context.rounding)
                        ans = ans._fix(context)
                        return ans
                    if not other:
                        exp = max(exp, self._exp - context.prec - 1)
                        ans = self._rescale(exp, context.rounding)
                        ans = ans._fix(context)
                        return ans
            op1 = _WorkRep(self)
            op2 = _WorkRep(other)
            op1, op2 = _normalize(op1, op2, context.prec)
            result = _WorkRep()
        if op1.sign != op2.sign:
            if op1.int == op2.int:
                ans = _dec_from_triple(negativezero, "0", exp)
                ans = ans._fix(context)
                return ans
            if op1.int < op2.int:
                op1, op2 = op2, op1
            if op1.sign == 1:
                result.sign = 1
                op1.sign, op2.sign = op2.sign, op1.sign
            else:
                result.sign = 0
        else:
            if op1.sign == 1:
                result.sign = 1
                op1.sign, op2.sign = (0, 0)
            else:
                result.sign = 0
            if op2.sign == 0:
                result.int = op1.int + op2.int
            else:
                result.int = op1.int - op2.int
            result.exp = op1.exp
            ans = Decimal(result)
            ans = ans._fix(context)
            return ans

    __radd__ = __add__

    def __sub__(self, other, context=None):
        """Return self - other"""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            if self._is_special or other._is_special:
                ans = self._check_nans(other, context=context)
                if ans:
                    return ans
            return self.__add__(other.copy_negate(), context=context)

    def __rsub__(self, other, context=None):
        """Return other - self"""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            return other.__sub__(self, context=context)

    def __mul__(self, other, context=None):
        """Return self * other.
        
        (+-) INF * 0 (or its reverse) raise InvalidOperation.
        """
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            if context is None:
                context = getcontext()
            resultsign = self._sign ^ other._sign
            if self._is_special or other._is_special:
                ans = self._check_nans(other, context)
                if ans:
                    return ans
                if self._isinfinity():
                    if not other:
                        return context._raise_error(InvalidOperation, "(+-)INF * 0")
                    return _SignedInfinity[resultsign]
                if other._isinfinity():
                    if not self:
                        return context._raise_error(InvalidOperation, "0 * (+-)INF")
                    return _SignedInfinity[resultsign]
                resultexp = self._exp + other._exp
                if not self or not other:
                    ans = _dec_from_triple(resultsign, "0", resultexp)
                    ans = ans._fix(context)
                    return ans
                if self._int == "1":
                    ans = _dec_from_triple(resultsign, other._int, resultexp)
                    ans = ans._fix(context)
                    return ans
                if other._int == "1":
                    ans = _dec_from_triple(resultsign, self._int, resultexp)
                    ans = ans._fix(context)
                    return ans
            op1 = _WorkRep(self)
            op2 = _WorkRep(other)
            ans = _dec_from_triple(resultsign, str(op1.int * op2.int), resultexp)
            ans = ans._fix(context)
            return ans

    __rmul__ = __mul__

    def __truediv__(self, other, context=None):
        """Return self / other."""
        other = _convert_other(other)
        if other is NotImplemented:
            return NotImplemented
        if context is None:
            context = getcontext()
        sign = self._sign ^ other._sign
        if self._is_special or other._is_special:
            ans = self._check_nans(other, context)
            if ans:
                return ans
            if self._isinfinity():
                if other._isinfinity():
                    return context._raise_error(InvalidOperation, "(+-)INF/(+-)INF")
                if self._isinfinity():
                    return _SignedInfinity[sign]
                if other._isinfinity():
                    context._raise_error(Clamped, "Division by infinity")
                    return _dec_from_triple(sign, "0", context.Etiny())
                if not other:
                    if not self:
                        return context._raise_error(DivisionUndefined, "0 / 0")
                    return context._raise_error(DivisionByZero, "x / 0", sign)
        if not self:
            exp = self._exp - other._exp
            coeff = 0
        else:
            shift = len(other._int) - len(self._int) + context.prec + 1
            exp = self._exp - other._exp - shift
            op1 = _WorkRep(self)
            op2 = _WorkRep(other)
            if shift >= 0:
                coeff, remainder = divmod(op1.int * 10 ** shift, op2.int)
            else:
                coeff, remainder = divmod(op1.int, op2.int * 10 ** (-shift))
        if remainder:
            if coeff % 5 == 0:
                coeff += 1
            else:
                ideal_exp = self._exp - other._exp
                while exp < ideal_exp:
                    if coeff % 10 == 0:
                        coeff //= 10
                        exp += 1

            ans = _dec_from_triple(sign, str(coeff), exp)
            return ans._fix(context)

    def _divide(self, other, context):
        """Return (self // other, self % other), to context.prec precision.
        
        Assumes that neither self nor other is a NaN, that self is not
        infinite and that other is nonzero.
        """
        sign = self._sign ^ other._sign
        if other._isinfinity():
            ideal_exp = self._exp
        else:
            ideal_exp = min(self._exp, other._exp)
        expdiff = self.adjusted() - other.adjusted()
        if not self or other._isinfinity() or expdiff <= -2:
            return (
                _dec_from_triple(sign, "0", 0),
                self._rescale(ideal_exp, context.rounding),
            )
        else:
            if expdiff <= context.prec:
                op1 = _WorkRep(self)
                op2 = _WorkRep(other)
                if op1.exp >= op2.exp:
                    op1.int *= 10 ** (op1.exp - op2.exp)
                else:
                    op2.int *= 10 ** (op2.exp - op1.exp)
                q, r = divmod(op1.int, op2.int)
                if q < 10 ** context.prec:
                    return (
                        _dec_from_triple(sign, str(q), 0),
                        _dec_from_triple(self._sign, str(r), ideal_exp),
                    )
                ans = context._raise_error(
                    DivisionImpossible, "quotient too large in //, % or divmod"
                )
            return (ans, ans)

    def __rtruediv__(self, other, context=None):
        """Swaps self/other and returns __truediv__."""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            return other.__truediv__(self, context=context)

    def __divmod__(self, other, context=None):
        """
        Return (self // other, self % other)
        """
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            if context is None:
                context = getcontext()
            ans = self._check_nans(other, context)
            if ans:
                return (ans, ans)
            sign = self._sign ^ other._sign
            if self._isinfinity():
                if other._isinfinity():
                    ans = context._raise_error(InvalidOperation, "divmod(INF, INF)")
                    return (ans, ans)
                return (
                    _SignedInfinity[sign],
                    context._raise_error(InvalidOperation, "INF % x"),
                )
            if not other:
                if not self:
                    ans = context._raise_error(DivisionUndefined, "divmod(0, 0)")
                    return (ans, ans)
                return (
                    context._raise_error(DivisionByZero, "x // 0", sign),
                    context._raise_error(InvalidOperation, "x % 0"),
                )
            quotient, remainder = self._divide(other, context)
            remainder = remainder._fix(context)
            return (quotient, remainder)

    def __rdivmod__(self, other, context=None):
        """Swaps self/other and returns __divmod__."""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            return other.__divmod__(self, context=context)

    def __mod__(self, other, context=None):
        """
        self % other
        """
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            if context is None:
                context = getcontext()
            ans = self._check_nans(other, context)
            if ans:
                return ans
            if self._isinfinity():
                return context._raise_error(InvalidOperation, "INF % x")
            if not other:
                if self:
                    return context._raise_error(InvalidOperation, "x % 0")
                return context._raise_error(DivisionUndefined, "0 % 0")
            remainder = self._divide(other, context)[1]
            remainder = remainder._fix(context)
            return remainder

    def __rmod__(self, other, context=None):
        """Swaps self/other and returns __mod__."""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            return other.__mod__(self, context=context)

    def remainder_near(self, other, context=None):
        """
        Remainder nearest to 0-  abs(remainder-near) <= other/2
        """
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        ans = self._check_nans(other, context)
        if ans:
            return ans
        elif self._isinfinity():
            return context._raise_error(InvalidOperation, "remainder_near(infinity, x)")
        elif not other:
            if self:
                return context._raise_error(InvalidOperation, "remainder_near(x, 0)")
            return context._raise_error(DivisionUndefined, "remainder_near(0, 0)")
        elif other._isinfinity():
            ans = Decimal(self)
            return ans._fix(context)
        else:
            ideal_exponent = min(self._exp, other._exp)
            if not self:
                ans = _dec_from_triple(self._sign, "0", ideal_exponent)
                return ans._fix(context)
            expdiff = self.adjusted() - other.adjusted()
            if expdiff >= context.prec + 1:
                return context._raise_error(DivisionImpossible)
            elif expdiff <= -2:
                ans = self._rescale(ideal_exponent, context.rounding)
                return ans._fix(context)
            op1 = _WorkRep(self)
            op2 = _WorkRep(other)
            if op1.exp >= op2.exp:
                op1.int *= 10 ** (op1.exp - op2.exp)
            else:
                op2.int *= 10 ** (op2.exp - op1.exp)
            q, r = divmod(op1.int, op2.int)
            if 2 * r + (q & 1) > op2.int:
                r -= op2.int
                q += 1
            if q >= 10 ** context.prec:
                return context._raise_error(DivisionImpossible)
            sign = self._sign
            if r < 0:
                sign = 1 - sign
                r = -r
            ans = _dec_from_triple(sign, str(r), ideal_exponent)
            return ans._fix(context)

    def __floordiv__(self, other, context=None):
        """self // other"""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        if context is None:
            context = getcontext()
        ans = self._check_nans(other, context)
        if ans:
            return ans
        elif self._isinfinity():
            if other._isinfinity():
                return context._raise_error(InvalidOperation, "INF // INF")
            return _SignedInfinity[self._sign ^ other._sign]
        elif not other:
            if self:
                return context._raise_error(
                    DivisionByZero, "x // 0", self._sign ^ other._sign
                )
            return context._raise_error(DivisionUndefined, "0 // 0")
        else:
            return self._divide(other, context)[0]

    def __rfloordiv__(self, other, context=None):
        """Swaps self/other and returns __floordiv__."""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            return other.__floordiv__(self, context=context)

    def __float__(self):
        """Float representation."""
        if self._isnan():
            if self.is_snan():
                raise ValueError("Cannot convert signaling NaN to float")
            s = "-nan" if self._sign else "nan"
        else:
            s = str(self)
        return float(s)

    def __int__(self):
        """Converts self to an int, truncating if necessary."""
        if self._is_special:
            if self._isnan():
                raise ValueError("Cannot convert NaN to integer")
            else:
                if self._isinfinity():
                    raise OverflowError("Cannot convert infinity to integer")
            s = (-1) ** self._sign
            if self._exp >= 0:
                return s * int(self._int) * 10 ** self._exp
            return s * int(self._int[: self._exp] or "0")

    __trunc__ = __int__

    def real(self):
        return self

    real = property(real)

    def imag(self):
        return Decimal(0)

    imag = property(imag)

    def conjugate(self):
        return self

    def __complex__(self):
        return complex(float(self))

    def _fix_nan(self, context):
        """Decapitate the payload of a NaN to fit the context"""
        payload = self._int
        max_payload_len = context.prec - context.clamp
        if len(payload) > max_payload_len:
            payload = payload[len(payload) - max_payload_len :].lstrip("0")
            return _dec_from_triple(self._sign, payload, self._exp, True)
        else:
            return Decimal(self)

    def _fix(self, context):
        """Round if it is necessary to keep self within prec precision.
        
        Rounds and fixes the exponent.  Does not raise on a sNaN.
        
        Arguments:
        self - Decimal instance
        context - context used.
        """
        if self._is_special:
            if self._isnan():
                return self._fix_nan(context)
            return Decimal(self)
        else:
            Etiny = context.Etiny()
            Etop = context.Etop()
            if not self:
                exp_max = [context.Emax, Etop][context.clamp]
                new_exp = min(max(self._exp, Etiny), exp_max)
                if new_exp != self._exp:
                    context._raise_error(Clamped)
                    return _dec_from_triple(self._sign, "0", new_exp)
                return Decimal(self)
            exp_min = len(self._int) + self._exp - context.prec
            if exp_min > Etop:
                ans = context._raise_error(Overflow, "above Emax", self._sign)
                context._raise_error(Inexact)
                context._raise_error(Rounded)
                return ans
            self_is_subnormal = exp_min < Etiny
            if self_is_subnormal:
                exp_min = Etiny
            if self._exp < exp_min:
                digits = len(self._int) + self._exp - exp_min
                if digits < 0:
                    self = _dec_from_triple(self._sign, "1", exp_min - 1)
                    digits = 0
                rounding_method = self._pick_rounding_function[context.rounding]
                changed = rounding_method(self, digits)
                coeff = self._int[:digits] or "0"
                if changed > 0:
                    coeff = str(int(coeff) + 1)
                    if len(coeff) > context.prec:
                        coeff = coeff[:-1]
                        exp_min += 1
                if exp_min > Etop:
                    ans = context._raise_error(Overflow, "above Emax", self._sign)
                else:
                    ans = _dec_from_triple(self._sign, coeff, exp_min)
                if changed:
                    if self_is_subnormal:
                        context._raise_error(Underflow)
                if self_is_subnormal:
                    context._raise_error(Subnormal)
                if changed:
                    context._raise_error(Inexact)
                context._raise_error(Rounded)
                if not ans:
                    context._raise_error(Clamped)
                return ans
            if self_is_subnormal:
                context._raise_error(Subnormal)
            if context.clamp == 1:
                if self._exp > Etop:
                    context._raise_error(Clamped)
                    self_padded = self._int + "0" * (self._exp - Etop)
                    return _dec_from_triple(self._sign, self_padded, Etop)
            return Decimal(self)

    def _round_down(self, prec):
        """Also known as round-towards-0, truncate."""
        if _all_zeros(self._int, prec):
            return 0
        else:
            return -1

    def _round_up(self, prec):
        """Rounds away from 0."""
        return -self._round_down(prec)

    def _round_half_up(self, prec):
        """Rounds 5 up (away from 0)"""
        if self._int[prec] in "56789":
            return 1
        elif _all_zeros(self._int, prec):
            return 0
        else:
            return -1

    def _round_half_down(self, prec):
        """Round 5 down"""
        if _exact_half(self._int, prec):
            return -1
        else:
            return self._round_half_up(prec)

    def _round_half_even(self, prec):
        """Round 5 to even, rest to nearest."""
        if _exact_half(self._int, prec):
            if prec == 0 or self._int[prec - 1] in "02468":
                return -1
            return self._round_half_up(prec)

    def _round_ceiling(self, prec):
        """Rounds up (not away from 0 if negative.)"""
        if self._sign:
            return self._round_down(prec)
        else:
            return -self._round_down(prec)

    def _round_floor(self, prec):
        """Rounds down (not towards 0 if negative)"""
        if not self._sign:
            return self._round_down(prec)
        else:
            return -self._round_down(prec)

    def _round_05up(self, prec):
        """Round down unless digit prec-1 is 0 or 5."""
        if prec:
            if self._int[prec - 1] not in "05":
                return self._round_down(prec)
            return -self._round_down(prec)

    _pick_rounding_function = dict(
        ROUND_DOWN=_round_down,
        ROUND_UP=_round_up,
        ROUND_HALF_UP=_round_half_up,
        ROUND_HALF_DOWN=_round_half_down,
        ROUND_HALF_EVEN=_round_half_even,
        ROUND_CEILING=_round_ceiling,
        ROUND_FLOOR=_round_floor,
        ROUND_05UP=_round_05up,
    )

    def __round__(self, n=None):
        """Round self to the nearest integer, or to a given precision.
        
        If only one argument is supplied, round a finite Decimal
        instance self to the nearest integer.  If self is infinite or
        a NaN then a Python exception is raised.  If self is finite
        and lies exactly halfway between two integers then it is
        rounded to the integer with even last digit.
        
        >>> round(Decimal('123.456'))
        123
        >>> round(Decimal('-456.789'))
        -457
        >>> round(Decimal('-3.0'))
        -3
        >>> round(Decimal('2.5'))
        2
        >>> round(Decimal('3.5'))
        4
        >>> round(Decimal('Inf'))
        Traceback (most recent call last):
          ...
        OverflowError: cannot round an infinity
        >>> round(Decimal('NaN'))
        Traceback (most recent call last):
          ...
        ValueError: cannot round a NaN
        
        If a second argument n is supplied, self is rounded to n
        decimal places using the rounding mode for the current
        context.
        
        For an integer n, round(self, -n) is exactly equivalent to
        self.quantize(Decimal('1En')).
        
        >>> round(Decimal('123.456'), 0)
        Decimal('123')
        >>> round(Decimal('123.456'), 2)
        Decimal('123.46')
        >>> round(Decimal('123.456'), -2)
        Decimal('1E+2')
        >>> round(Decimal('-Infinity'), 37)
        Decimal('NaN')
        >>> round(Decimal('sNaN123'), 0)
        Decimal('NaN123')
        
        """
        if n is not None:
            if not isinstance(n, int):
                raise TypeError("Second argument to round should be integral")
            exp = _dec_from_triple(0, "1", -n)
            return self.quantize(exp)
        else:
            if self._is_special:
                if self.is_nan():
                    raise ValueError("cannot round a NaN")
                else:
                    raise OverflowError("cannot round an infinity")
            return int(self._rescale(0, ROUND_HALF_EVEN))

    def __floor__(self):
        """Return the floor of self, as an integer.
        
        For a finite Decimal instance self, return the greatest
        integer n such that n <= self.  If self is infinite or a NaN
        then a Python exception is raised.
        
        """
        if self._is_special:
            if self.is_nan():
                raise ValueError("cannot round a NaN")
            else:
                raise OverflowError("cannot round an infinity")
            return int(self._rescale(0, ROUND_FLOOR))

    def __ceil__(self):
        """Return the ceiling of self, as an integer.
        
        For a finite Decimal instance self, return the least integer n
        such that n >= self.  If self is infinite or a NaN then a
        Python exception is raised.
        
        """
        if self._is_special:
            if self.is_nan():
                raise ValueError("cannot round a NaN")
            else:
                raise OverflowError("cannot round an infinity")
            return int(self._rescale(0, ROUND_CEILING))

    def fma(self, other, third, context=None):
        """Fused multiply-add.
        
        Returns self*other+third with no rounding of the intermediate
        product self*other.
        
        self and other are multiplied together, with no rounding of
        the result.  The third operand is then added to the result,
        and a single final rounding is performed.
        """
        other = _convert_other(other, raiseit=True)
        third = _convert_other(third, raiseit=True)
        if self._is_special or other._is_special:
            if context is None:
                context = getcontext()
            if self._exp == "N":
                return context._raise_error(InvalidOperation, "sNaN", self)
            if other._exp == "N":
                return context._raise_error(InvalidOperation, "sNaN", other)
            if self._exp == "n":
                product = self
            else:
                if other._exp == "n":
                    product = other
                else:
                    if self._exp == "F":
                        if not other:
                            return context._raise_error(
                                InvalidOperation, "INF * 0 in fma"
                            )
                        product = _SignedInfinity[self._sign ^ other._sign]
                    else:
                        if other._exp == "F":
                            if not self:
                                return context._raise_error(
                                    InvalidOperation, "0 * INF in fma"
                                )
                            product = _SignedInfinity[self._sign ^ other._sign]
        else:
            product = _dec_from_triple(
                self._sign ^ other._sign,
                str(int(self._int) * int(other._int)),
                self._exp + other._exp,
            )
        return product.__add__(third, context)

    def _power_modulo(self, other, modulo, context=None):
        """Three argument version of __pow__"""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            modulo = _convert_other(modulo)
            if modulo is NotImplemented:
                return modulo
            if context is None:
                context = getcontext()
            self_is_nan = self._isnan()
            other_is_nan = other._isnan()
            modulo_is_nan = modulo._isnan()
            if self_is_nan or other_is_nan or modulo_is_nan:
                if self_is_nan == 2:
                    return context._raise_error(InvalidOperation, "sNaN", self)
                if other_is_nan == 2:
                    return context._raise_error(InvalidOperation, "sNaN", other)
                if modulo_is_nan == 2:
                    return context._raise_error(InvalidOperation, "sNaN", modulo)
                if self_is_nan:
                    return self._fix_nan(context)
                if other_is_nan:
                    return other._fix_nan(context)
                return modulo._fix_nan(context)
            if not (self._isinteger() and other._isinteger() and modulo._isinteger()):
                return context._raise_error(
                    InvalidOperation,
                    "pow() 3rd argument not allowed unless all arguments are integers",
                )
            if other < 0:
                return context._raise_error(
                    InvalidOperation,
                    "pow() 2nd argument cannot be negative when 3rd argument specified",
                )
            if not modulo:
                return context._raise_error(
                    InvalidOperation, "pow() 3rd argument cannot be 0"
                )
            if modulo.adjusted() >= context.prec:
                return context._raise_error(
                    InvalidOperation,
                    "insufficient precision: pow() 3rd argument must not have more than precision digits",
                )
            if not other:
                if not self:
                    return context._raise_error(
                        InvalidOperation,
                        "at least one of pow() 1st argument and 2nd argument must be nonzero ;0**0 is not defined",
                    )
                if other._iseven():
                    sign = 0
                else:
                    sign = self._sign
                modulo = abs(int(modulo))
                base = _WorkRep(self.to_integral_value())
                exponent = _WorkRep(other.to_integral_value())
                base = base.int % modulo * pow(10, base.exp, modulo) % modulo
                for i in range(exponent.exp):
                    base = pow(base, 10, modulo)

            base = pow(base, exponent.int, modulo)
            return _dec_from_triple(sign, str(base), 0)

    def _power_exact(self, other, p):
        """Attempt to compute self**other exactly.
        
        Given Decimals self and other and an integer p, attempt to
        compute an exact result for the power self**other, with p
        digits of precision.  Return None if self**other is not
        exactly representable in p digits.
        
        Assumes that elimination of special cases has already been
        performed: self and other must both be nonspecial; self must
        be positive and not numerically equal to 1; other must be
        nonzero.  For efficiency, other._exp should not be too large,
        so that 10**abs(other._exp) is a feasible calculation."""
        x = _WorkRep(self)
        xc, xe = x.int, x.exp
        while xc % 10 == 0:
            xc //= 10
            xe += 1

        y = _WorkRep(other)
        yc, ye = y.int, y.exp
        while yc % 10 == 0:
            yc //= 10
            ye += 1

        if xc == 1:
            xe *= yc
            while xe % 10 == 0:
                xe //= 10
                ye += 1

            if ye < 0:
                return
            exponent = xe * 10 ** ye
            if y.sign == 1:
                exponent = -exponent
            if other._isinteger() and other._sign == 0:
                ideal_exponent = self._exp * int(other)
                zeros = min(exponent - ideal_exponent, p - 1)
            else:
                zeros = 0
            return _dec_from_triple(0, "1" + "0" * zeros, exponent - zeros)
        if y.sign == 1:
            last_digit = xc % 10
            if last_digit in (2, 4, 6, 8):
                if xc & -xc != xc:
                    return
                e = _nbits(xc) - 1
                emax = p * 93 // 65
                if ye >= len(str(emax)):
                    return
                e = _decimal_lshift_exact(e * yc, ye)
                xe = _decimal_lshift_exact(xe * yc, ye)
                if e is None or xe is None:
                    return
                if e > emax:
                    return
                xc = 5 ** e
            else:
                if last_digit == 5:
                    e = _nbits(xc) * 28 // 65
                    xc, remainder = divmod(5 ** e, xc)
                    if remainder:
                        return
                    while xc % 5 == 0:
                        xc //= 5
                        e -= 1

                    emax = p * 10 // 3
                    if ye >= len(str(emax)):
                        return
                    e = _decimal_lshift_exact(e * yc, ye)
                    xe = _decimal_lshift_exact(xe * yc, ye)
                    if e is None or xe is None:
                        return
                    if e > emax:
                        return
                    xc = 2 ** e
                else:
                    return
                if xc >= 10 ** p:
                    return
            xe = -e - xe
            return _dec_from_triple(0, str(xc), xe)
        if ye >= 0:
            m, n = yc * 10 ** ye, 1
        else:
            if xe != 0:
                if len(str(abs(yc * xe))) <= -ye:
                    return
                xc_bits = _nbits(xc)
                if xc != 1:
                    if len(str(abs(yc) * xc_bits)) <= -ye:
                        return
                    m, n = yc, 10 ** (-ye)
                    while m % 2 == n % 2 == 0:
                        m //= 2
                        n //= 2

                    while m % 5 == n % 5 == 0:
                        m //= 5
                        n //= 5

                if n > 1:
                    if xc != 1:
                        if xc_bits <= n:
                            return
                        xe, rem = divmod(xe, n)
                        if rem != 0:
                            return
                        a = 1 << -(-_nbits(xc) // n)
                        while 1:
                            q, r = divmod(xc, a ** (n - 1))
                            if a <= q:
                                break
                            else:
                                a = (a * (n - 1) + q) // n

                        if not (a == q and r == 0):
                            return
                        xc = a
                if xc > 1:
                    if m > p * 100 // _log10_lb(xc):
                        return
                    xc = xc ** m
                    xe *= m
                    if xc > 10 ** p:
                        return
                    str_xc = str(xc)
                    if other._isinteger():
                        if other._sign == 0:
                            ideal_exponent = self._exp * int(other)
                            zeros = min(xe - ideal_exponent, p - len(str_xc))
                        zeros = 0
                    return _dec_from_triple(0, str_xc + "0" * zeros, xe - zeros)

    def __pow__(self, other, modulo=None, context=None):
        """Return self ** other [ % modulo].
        
        With two arguments, compute self**other.
        
        With three arguments, compute (self**other) % modulo.  For the
        three argument form, the following restrictions on the
        arguments hold:
        
         - all three arguments must be integral
         - other must be nonnegative
         - either self or other (or both) must be nonzero
         - modulo must be nonzero and must have at most p digits,
           where p is the context precision.
        
        If any of these restrictions is violated the InvalidOperation
        flag is raised.
        
        The result of pow(self, other, modulo) is identical to the
        result that would be obtained by computing (self**other) %
        modulo with unbounded precision, but is computed more
        efficiently.  It is always exact.
        """
        if modulo is not None:
            return self._power_modulo(other, modulo, context)
        else:
            other = _convert_other(other)
            if other is NotImplemented:
                return other
            if context is None:
                context = getcontext()
            ans = self._check_nans(other, context)
            if ans:
                return ans
            if not other:
                if not self:
                    return context._raise_error(InvalidOperation, "0 ** 0")
                return _One
            result_sign = 0
            if self._sign == 1:
                if other._isinteger():
                    if not other._iseven():
                        result_sign = 1
                    else:
                        if self:
                            return context._raise_error(
                                InvalidOperation,
                                "x ** y with x negative and y not an integer",
                            )
                    self = self.copy_negate()
                if not self:
                    if other._sign == 0:
                        return _dec_from_triple(result_sign, "0", 0)
                    return _SignedInfinity[result_sign]
                if self._isinfinity():
                    if other._sign == 0:
                        return _SignedInfinity[result_sign]
                    return _dec_from_triple(result_sign, "0", 0)
            if self == _One:
                if other._isinteger():
                    if other._sign == 1:
                        multiplier = 0
                    else:
                        if other > context.prec:
                            multiplier = context.prec
                        else:
                            multiplier = int(other)
                        exp = self._exp * multiplier
                        if exp < 1 - context.prec:
                            exp = 1 - context.prec
                            context._raise_error(Rounded)
                        else:
                            context._raise_error(Inexact)
                            context._raise_error(Rounded)
                            exp = 1 - context.prec
                    return _dec_from_triple(result_sign, "1" + "0" * -exp, exp)
                self_adj = self.adjusted()
                if other._isinfinity():
                    if (other._sign == 0) == (self_adj < 0):
                        return _dec_from_triple(result_sign, "0", 0)
                    return _SignedInfinity[result_sign]
                ans = None
                exact = False
                bound = self._log10_exp_bound() + other.adjusted()
                if (self_adj >= 0) == (other._sign == 0):
                    if bound >= len(str(context.Emax)):
                        ans = _dec_from_triple(result_sign, "1", context.Emax + 1)
                    else:
                        Etiny = context.Etiny()
                        if bound >= len(str(-Etiny)):
                            ans = _dec_from_triple(result_sign, "1", Etiny - 1)
                    if ans is None:
                        ans = self._power_exact(other, context.prec + 1)
                        if ans is not None:
                            if result_sign == 1:
                                ans = _dec_from_triple(1, ans._int, ans._exp)
                            exact = True
                    if ans is None:
                        p = context.prec
                        x = _WorkRep(self)
                        xc, xe = x.int, x.exp
                        y = _WorkRep(other)
                        yc, ye = y.int, y.exp
                        if y.sign == 1:
                            yc = -yc
                        extra = 3
                        while True:
                            coeff, exp = _dpower(xc, xe, yc, ye, p + extra)
                            if coeff % (5 * 10 ** (len(str(coeff)) - p - 1)):
                                break
                            extra += 3

                        ans = _dec_from_triple(result_sign, str(coeff), exp)
            if exact:
                if not other._isinteger():
                    if len(ans._int) <= context.prec:
                        expdiff = context.prec + 1 - len(ans._int)
                        ans = _dec_from_triple(
                            ans._sign, ans._int + "0" * expdiff, ans._exp - expdiff
                        )
                    newcontext = context.copy()
                    newcontext.clear_flags()
                    for exception in _signals:
                        newcontext.traps[exception] = 0

                    ans = ans._fix(newcontext)
                    newcontext._raise_error(Inexact)
                    if newcontext.flags[Subnormal]:
                        newcontext._raise_error(Underflow)
                    if newcontext.flags[Overflow]:
                        context._raise_error(Overflow, "above Emax", ans._sign)
                    for exception in (Underflow, Subnormal, Inexact, Rounded, Clamped):
                        if newcontext.flags[exception]:
                            context._raise_error(exception)

                ans = ans._fix(context)
            return ans

    def __rpow__(self, other, context=None):
        """Swaps self/other and returns __pow__."""
        other = _convert_other(other)
        if other is NotImplemented:
            return other
        else:
            return other.__pow__(self, context=context)

    def normalize(self, context=None):
        """Normalize- strip trailing 0s, change anything equal to 0 to 0e0"""
        if context is None:
            context = getcontext()
        ans = (self._is_special and self)._check_nans(context=context)
        if ans:
            return ans
        else:
            dup = self._fix(context)
            if dup._isinfinity():
                return dup
            if not dup:
                return _dec_from_triple(dup._sign, "0", 0)
            exp_max = [context.Emax, context.Etop()][context.clamp]
            end = len(dup._int)
            exp = dup._exp
            while dup._int[end - 1] == "0":
                if exp < exp_max:
                    exp += 1
                    end -= 1

            return _dec_from_triple(dup._sign, dup._int[:end], exp)

    def quantize(self, exp, rounding=None, context=None):
        """Quantize self so its exponent is the same as that of exp.
        
        Similar to self._rescale(exp._exp) but with error checking.
        """
        exp = _convert_other(exp, raiseit=True)
        if context is None:
            context = getcontext()
        if rounding is None:
            rounding = context.rounding
        ans = (self._is_special or exp._is_special) and self._check_nans(exp, context)
        if ans:
            return ans
        else:
            if exp._isinfinity() or self._isinfinity():
                if exp._isinfinity():
                    if self._isinfinity():
                        return Decimal(self)
                    return context._raise_error(
                        InvalidOperation, "quantize with one INF"
                    )
                if not context.Etiny() <= exp._exp <= context.Emax:
                    return context._raise_error(
                        InvalidOperation, "target exponent out of bounds in quantize"
                    )
                if not self:
                    ans = _dec_from_triple(self._sign, "0", exp._exp)
                    return ans._fix(context)
                self_adjusted = self.adjusted()
                if self_adjusted > context.Emax:
                    return context._raise_error(
                        InvalidOperation,
                        "exponent of quantize result too large for current context",
                    )
                if self_adjusted - exp._exp + 1 > context.prec:
                    return context._raise_error(
                        InvalidOperation,
                        "quantize result has too many digits for current context",
                    )
                ans = self._rescale(exp._exp, rounding)
                if ans.adjusted() > context.Emax:
                    return context._raise_error(
                        InvalidOperation,
                        "exponent of quantize result too large for current context",
                    )
                if len(ans._int) > context.prec:
                    return context._raise_error(
                        InvalidOperation,
                        "quantize result has too many digits for current context",
                    )
                if ans:
                    if ans.adjusted() < context.Emin:
                        context._raise_error(Subnormal)
                if ans._exp > self._exp:
                    if ans != self:
                        context._raise_error(Inexact)
                    context._raise_error(Rounded)
                ans = ans._fix(context)
            return ans

    def same_quantum(self, other, context=None):
        """Return True if self and other have the same exponent; otherwise
        return False.
        
        If either operand is a special value, the following rules are used:
           * return True if both operands are infinities
           * return True if both operands are NaNs
           * otherwise, return False.
        """
        other = _convert_other(other, raiseit=True)
        if self._is_special or other._is_special:
            return (
                self.is_nan()
                and other.is_nan()
                or self.is_infinite()
                and other.is_infinite()
            )
        else:
            return self._exp == other._exp

    def _rescale(self, exp, rounding):
        """Rescale self so that the exponent is exp, either by padding with zeros
        or by truncating digits, using the given rounding mode.
        
        Specials are returned without change.  This operation is
        quiet: it raises no flags, and uses no information from the
        context.
        
        exp = exp to scale to (an integer)
        rounding = rounding mode
        """
        if self._is_special:
            return Decimal(self)
        elif not self:
            return _dec_from_triple(self._sign, "0", exp)
        elif self._exp >= exp:
            return _dec_from_triple(
                self._sign, self._int + "0" * (self._exp - exp), exp
            )
        else:
            digits = len(self._int) + self._exp - exp
            if digits < 0:
                self = _dec_from_triple(self._sign, "1", exp - 1)
                digits = 0
            this_function = self._pick_rounding_function[rounding]
            changed = this_function(self, digits)
            coeff = self._int[:digits] or "0"
            if changed == 1:
                coeff = str(int(coeff) + 1)
            return _dec_from_triple(self._sign, coeff, exp)

    def _round(self, places, rounding):
        """Round a nonzero, nonspecial Decimal to a fixed number of
        significant figures, using the given rounding mode.
        
        Infinities, NaNs and zeros are returned unaltered.
        
        This operation is quiet: it raises no flags, and uses no
        information from the context.
        
        """
        if places <= 0:
            raise ValueError("argument should be at least 1 in _round")
        if self._is_special or not self:
            return Decimal(self)
        else:
            ans = self._rescale(self.adjusted() + 1 - places, rounding)
            if ans.adjusted() != self.adjusted():
                ans = ans._rescale(ans.adjusted() + 1 - places, rounding)
            return ans

    def to_integral_exact(self, rounding=None, context=None):
        """Rounds to a nearby integer.
        
        If no rounding mode is specified, take the rounding mode from
        the context.  This method raises the Rounded and Inexact flags
        when appropriate.
        
        See also: to_integral_value, which does exactly the same as
        this method except that it doesn't raise Inexact or Rounded.
        """
        if self._is_special:
            ans = self._check_nans(context=context)
            if ans:
                return ans
            return Decimal(self)
        elif self._exp >= 0:
            return Decimal(self)
        elif not self:
            return _dec_from_triple(self._sign, "0", 0)
        else:
            if context is None:
                context = getcontext()
            if rounding is None:
                rounding = context.rounding
            ans = self._rescale(0, rounding)
            if ans != self:
                context._raise_error(Inexact)
            context._raise_error(Rounded)
            return ans

    def to_integral_value(self, rounding=None, context=None):
        """Rounds to the nearest integer, without raising inexact, rounded."""
        if context is None:
            context = getcontext()
        if rounding is None:
            rounding = context.rounding
        if self._is_special:
            ans = self._check_nans(context=context)
            if ans:
                return ans
            return Decimal(self)
        elif self._exp >= 0:
            return Decimal(self)
        else:
            return self._rescale(0, rounding)

    to_integral = to_integral_value

    def sqrt(self, context=None):
        """Return the square root of self."""
        if context is None:
            context = getcontext()
        ans = (self._is_special and self)._check_nans(context=context)
        if ans:
            return ans
        else:
            if self._isinfinity():
                if self._sign == 0:
                    return Decimal(self)
                if not self:
                    ans = _dec_from_triple(self._sign, "0", self._exp // 2)
                    return ans._fix(context)
                if self._sign == 1:
                    return context._raise_error(InvalidOperation, "sqrt(-x), x > 0")
                prec = context.prec + 1
                op = _WorkRep(self)
                e = op.exp >> 1
                if op.exp & 1:
                    c = op.int * 10
                    l = (len(self._int) >> 1) + 1
                else:
                    c = op.int
                    l = len(self._int) + 1 >> 1
                shift = prec - l
                if shift >= 0:
                    c *= 100 ** shift
                    exact = True
                else:
                    c, remainder = divmod(c, 100 ** (-shift))
                    exact = not remainder
                e -= shift
                n = 10 ** prec
                while 1:
                    q = c // n
                    if n <= q:
                        break
                    else:
                        n = n + q >> 1

                exact = exact and n * n == c
                if exact:
                    if shift >= 0:
                        n //= 10 ** shift
                    else:
                        n *= 10 ** (-shift)
                    e += shift
                else:
                    if n % 5 == 0:
                        n += 1
                ans = _dec_from_triple(0, str(n), e)
                context = context._shallow_copy()
                rounding = context._set_rounding(ROUND_HALF_EVEN)
                ans = ans._fix(context)
                context.rounding = rounding
            return ans

    def max(self, other, context=None):
        """Returns the larger value.
        
        Like max(self, other) except if one is not a number, returns
        NaN (and signals if one is sNaN).  Also rounds.
        """
        other = _convert_other(other, raiseit=True)
        if context is None:
            context = getcontext()
        if self._is_special or other._is_special:
            sn = self._isnan()
            on = other._isnan()
            if sn or on:
                if on == 1:
                    pass
        if sn == 0:
            return self._fix(context)
        elif sn == 1:
            if on == 0:
                return other._fix(context)
            return self._check_nans(other, context)
        else:
            c = self._cmp(other)
            if c == 0:
                c = self.compare_total(other)
            if c == -1:
                ans = other
            else:
                ans = self
            return ans._fix(context)

    def min(self, other, context=None):
        """Returns the smaller value.
        
        Like min(self, other) except if one is not a number, returns
        NaN (and signals if one is sNaN).  Also rounds.
        """
        other = _convert_other(other, raiseit=True)
        if context is None:
            context = getcontext()
        if self._is_special or other._is_special:
            sn = self._isnan()
            on = other._isnan()
            if sn or on:
                if on == 1:
                    pass
        if sn == 0:
            return self._fix(context)
        elif sn == 1:
            if on == 0:
                return other._fix(context)
            return self._check_nans(other, context)
        else:
            c = self._cmp(other)
            if c == 0:
                c = self.compare_total(other)
            if c == -1:
                ans = self
            else:
                ans = other
            return ans._fix(context)

    def _isinteger(self):
        """Returns whether self is an integer"""
        if self._is_special:
            return False
        elif self._exp >= 0:
            return True
        else:
            rest = self._int[self._exp :]
            return rest == "0" * len(rest)

    def _iseven(self):
        """Returns True if self is even.  Assumes self is an integer."""
        if not self or self._exp > 0:
            return True
        else:
            return self._int[-1 + self._exp] in "02468"

    def adjusted(self):
        """Return the adjusted exponent of self"""
        try:
            return self._exp + len(self._int) - 1
        except TypeError:
            return 0

    def canonical(self):
        """Returns the same Decimal object.
        
        As we do not have different encodings for the same number, the
        received object already is in its canonical form.
        """
        return self

    def compare_signal(self, other, context=None):
        """Compares self to the other operand numerically.
        
        It's pretty much like compare(), but all NaNs signal, with signaling
        NaNs taking precedence over quiet NaNs.
        """
        other = _convert_other(other, raiseit=True)
        ans = self._compare_check_nans(other, context)
        if ans:
            return ans
        else:
            return self.compare(other, context=context)

    def compare_total(self, other, context=None):
        """Compares self to other using the abstract representations.
        
        This is not like the standard compare, which use their numerical
        value. Note that a total ordering is defined for all possible abstract
        representations.
        """
        other = _convert_other(other, raiseit=True)
        if self._sign:
            if not other._sign:
                return _NegativeOne
            if not self._sign:
                if other._sign:
                    return _One
                sign = self._sign
                self_nan = self._isnan()
                other_nan = other._isnan()
                if self_nan or other_nan:
                    if self_nan == other_nan:
                        self_key = (len(self._int), self._int)
                        other_key = (len(other._int), other._int)
                        if self_key < other_key:
                            if sign:
                                return _One
                            return _NegativeOne
                        if self_key > other_key:
                            if sign:
                                return _NegativeOne
                            return _One
                        return _Zero
        if sign:
            if self_nan == 1:
                return _NegativeOne
            if other_nan == 1:
                return _One
            if self_nan == 2:
                return _NegativeOne
            if other_nan == 2:
                return _One
        else:
            if self_nan == 1:
                return _One
            if other_nan == 1:
                return _NegativeOne
            if self_nan == 2:
                return _One
            if other_nan == 2:
                return _NegativeOne
            if self < other:
                return _NegativeOne
            if self > other:
                return _One
            if self._exp < other._exp:
                if sign:
                    return _One
                return _NegativeOne
            if self._exp > other._exp:
                if sign:
                    return _NegativeOne
                return _One
            return _Zero

    def compare_total_mag(self, other, context=None):
        """Compares self to other using abstract repr., ignoring sign.
        
        Like compare_total, but with operand's sign ignored and assumed to be 0.
        """
        other = _convert_other(other, raiseit=True)
        s = self.copy_abs()
        o = other.copy_abs()
        return s.compare_total(o)

    def copy_abs(self):
        """Returns a copy with the sign set to 0. """
        return _dec_from_triple(0, self._int, self._exp, self._is_special)

    def copy_negate(self):
        """Returns a copy with the sign inverted."""
        if self._sign:
            return _dec_from_triple(0, self._int, self._exp, self._is_special)
        else:
            return _dec_from_triple(1, self._int, self._exp, self._is_special)

    def copy_sign(self, other, context=None):
        """Returns self with the sign of other."""
        other = _convert_other(other, raiseit=True)
        return _dec_from_triple(other._sign, self._int, self._exp, self._is_special)

    def exp(self, context=None):
        """Returns e ** self."""
        if context is None:
            context = getcontext()
        ans = self._check_nans(context=context)
        if ans:
            return ans
        if self._isinfinity() == -1:
            return _Zero
        if not self:
            return _One
        if self._isinfinity() == 1:
            return Decimal(self)
        p = context.prec
        adj = self.adjusted()
        if self._sign == 0:
            pass
        if adj > len(str((context.Emax + 1) * 3)):
            ans = _dec_from_triple(0, "1", context.Emax + 1)
        if self._sign == 1:
            if adj > len(str((-context.Etiny() + 1) * 3)):
                ans = _dec_from_triple(0, "1", context.Etiny() - 1)
            if self._sign == 0 and adj < -p:
                ans = _dec_from_triple(0, "1" + "0" * (p - 1) + "1", -p)
            else:
                if self._sign == 1:
                    if adj < -p - 1:
                        ans = _dec_from_triple(0, "9" * (p + 1), -p - 1)
                    op = _WorkRep(self)
                    c, e = op.int, op.exp
                    if op.sign == 1:
                        c = -c
                    extra = 3
                    while True:
                        coeff, exp = _dexp(c, e, p + extra)
                        if coeff % (5 * 10 ** (len(str(coeff)) - p - 1)):
                            break
                        extra += 3

                    ans = _dec_from_triple(0, str(coeff), exp)
            context = context._shallow_copy()
            rounding = context._set_rounding(ROUND_HALF_EVEN)
            ans = ans._fix(context)
            context.rounding = rounding
            return ans

    def is_canonical(self):
        """Return True if self is canonical; otherwise return False.
        
        Currently, the encoding of a Decimal instance is always
        canonical, so this method returns True for any Decimal.
        """
        return True

    def is_finite(self):
        """Return True if self is finite; otherwise return False.
        
        A Decimal instance is considered finite if it is neither
        infinite nor a NaN.
        """
        return not self._is_special

    def is_infinite(self):
        """Return True if self is infinite; otherwise return False."""
        return self._exp == "F"

    def is_nan(self):
        """Return True if self is a qNaN or sNaN; otherwise return False."""
        return self._exp in ("n", "N")

    def is_normal(self, context=None):
        """Return True if self is a normal number; otherwise return False."""
        if self._is_special or not self:
            return False
        else:
            if context is None:
                context = getcontext()
            return context.Emin <= self.adjusted()

    def is_qnan(self):
        """Return True if self is a quiet NaN; otherwise return False."""
        return self._exp == "n"

    def is_signed(self):
        """Return True if self is negative; otherwise return False."""
        return self._sign == 1

    def is_snan(self):
        """Return True if self is a signaling NaN; otherwise return False."""
        return self._exp == "N"

    def is_subnormal(self, context=None):
        """Return True if self is subnormal; otherwise return False."""
        if self._is_special or not self:
            return False
        else:
            if context is None:
                context = getcontext()
            return self.adjusted() < context.Emin

    def is_zero(self):
        """Return True if self is a zero; otherwise return False."""
        return not self._is_special and self._int == "0"

    def _ln_exp_bound(self):
        """Compute a lower bound for the adjusted exponent of self.ln().
        In other words, compute r such that self.ln() >= 10**r.  Assumes
        that self is finite and positive and that self != 1.
        """
        adj = self._exp + len(self._int) - 1
        if adj >= 1:
            return len(str(adj * 23 // 10)) - 1
        elif adj <= -2:
            return len(str((-1 - adj) * 23 // 10)) - 1
        op = _WorkRep(self)
        c, e = op.int, op.exp
        if adj == 0:
            num = str(c - 10 ** (-e))
            den = str(c)
            return len(num) - len(den) - (num < den)
        else:
            return e + len(str(10 ** (-e) - c)) - 1

    def ln(self, context=None):
        """Returns the natural (base e) logarithm of self."""
        if context is None:
            context = getcontext()
        ans = self._check_nans(context=context)
        if ans:
            return ans
        elif not self:
            return _NegativeInfinity
        elif self._isinfinity() == 1:
            return _Infinity
        elif self == _One:
            return _Zero
        elif self._sign == 1:
            return context._raise_error(InvalidOperation, "ln of a negative value")
        else:
            op = _WorkRep(self)
            c, e = op.int, op.exp
            p = context.prec
            places = p - self._ln_exp_bound() + 2
            while True:
                coeff = _dlog(c, e, places)
                if coeff % (5 * 10 ** (len(str(abs(coeff))) - p - 1)):
                    break
                places += 3

            ans = _dec_from_triple(int(coeff < 0), str(abs(coeff)), -places)
            context = context._shallow_copy()
            rounding = context._set_rounding(ROUND_HALF_EVEN)
            ans = ans._fix(context)
            context.rounding = rounding
            return ans

    def _log10_exp_bound(self):
        """Compute a lower bound for the adjusted exponent of self.log10().
        In other words, find r such that self.log10() >= 10**r.
        Assumes that self is finite and positive and that self != 1.
        """
        adj = self._exp + len(self._int) - 1
        if adj >= 1:
            return len(str(adj)) - 1
        elif adj <= -2:
            return len(str(-1 - adj)) - 1
        else:
            op = _WorkRep(self)
            c, e = op.int, op.exp
            if adj == 0:
                num = str(c - 10 ** (-e))
                den = str(231 * c)
                return len(num) - len(den) - (num < den) + 2
            num = str(10 ** (-e) - c)
            return len(num) + e - (num < "231") - 1

    def log10(self, context=None):
        """Returns the base 10 logarithm of self."""
        if context is None:
            context = getcontext()
        ans = self._check_nans(context=context)
        if ans:
            return ans
        elif not self:
            return _NegativeInfinity
        elif self._isinfinity() == 1:
            return _Infinity
        elif self._sign == 1:
            return context._raise_error(InvalidOperation, "log10 of a negative value")
        else:
            if self._int[0] == "1":
                if self._int[1:] == "0" * (len(self._int) - 1):
                    ans = Decimal(self._exp + len(self._int) - 1)
                op = _WorkRep(self)
                c, e = op.int, op.exp
                p = context.prec
                places = p - self._log10_exp_bound() + 2
                while True:
                    coeff = _dlog10(c, e, places)
                    if coeff % (5 * 10 ** (len(str(abs(coeff))) - p - 1)):
                        break
                    places += 3

                ans = _dec_from_triple(int(coeff < 0), str(abs(coeff)), -places)
            context = context._shallow_copy()
            rounding = context._set_rounding(ROUND_HALF_EVEN)
            ans = ans._fix(context)
            context.rounding = rounding
            return ans

    def logb(self, context=None):
        """ Returns the exponent of the magnitude of self's MSD.
        
        The result is the integer which is the exponent of the magnitude
        of the most significant digit of self (as though it were truncated
        to a single digit while maintaining the value of that digit and
        without limiting the resulting exponent).
        """
        ans = self._check_nans(context=context)
        if ans:
            return ans
        else:
            if context is None:
                context = getcontext()
            if self._isinfinity():
                return _Infinity
            if not self:
                return context._raise_error(DivisionByZero, "logb(0)", 1)
            ans = Decimal(self.adjusted())
            return ans._fix(context)

    def _islogical(self):
        """Return True if self is a logical operand.
        
        For being logical, it must be a finite number with a sign of 0,
        an exponent of 0, and a coefficient whose digits must all be
        either 0 or 1.
        """
        if self._sign != 0 or self._exp != 0:
            return False
        else:
            for dig in self._int:
                if dig not in "01":
                    return False

            return True

    def _fill_logical(self, context, opa, opb):
        dif = context.prec - len(opa)
        if dif > 0:
            opa = "0" * dif + opa
        else:
            if dif < 0:
                opa = opa[-context.prec :]
        dif = context.prec - len(opb)
        if dif > 0:
            opb = "0" * dif + opb
        else:
            if dif < 0:
                opb = opb[-context.prec :]
        return (opa, opb)

    def logical_and(self, other, context=None):
        """Applies an 'and' operation between self and other's digits."""
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        if not self._islogical() or not other._islogical():
            return context._raise_error(InvalidOperation)
        else:
            opa, opb = self._fill_logical(context, self._int, other._int)
            result = ("").join([str(int(a) & int(b)) for a, b in zip(opa, opb)])
            return _dec_from_triple(0, result.lstrip("0") or "0", 0)

    def logical_invert(self, context=None):
        """Invert all its digits."""
        if context is None:
            context = getcontext()
        return self.logical_xor(_dec_from_triple(0, "1" * context.prec, 0), context)

    def logical_or(self, other, context=None):
        """Applies an 'or' operation between self and other's digits."""
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        if not self._islogical() or not other._islogical():
            return context._raise_error(InvalidOperation)
        else:
            opa, opb = self._fill_logical(context, self._int, other._int)
            result = ("").join([str(int(a) | int(b)) for a, b in zip(opa, opb)])
            return _dec_from_triple(0, result.lstrip("0") or "0", 0)

    def logical_xor(self, other, context=None):
        """Applies an 'xor' operation between self and other's digits."""
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        if not self._islogical() or not other._islogical():
            return context._raise_error(InvalidOperation)
        else:
            opa, opb = self._fill_logical(context, self._int, other._int)
            result = ("").join([str(int(a) ^ int(b)) for a, b in zip(opa, opb)])
            return _dec_from_triple(0, result.lstrip("0") or "0", 0)

    def max_mag(self, other, context=None):
        """Compares the values numerically with their sign ignored."""
        other = _convert_other(other, raiseit=True)
        if context is None:
            context = getcontext()
        if self._is_special or other._is_special:
            sn = self._isnan()
            on = other._isnan()
            if sn or on:
                if on == 1:
                    pass
        if sn == 0:
            return self._fix(context)
        elif sn == 1:
            if on == 0:
                return other._fix(context)
            return self._check_nans(other, context)
        else:
            c = self.copy_abs()._cmp(other.copy_abs())
            if c == 0:
                c = self.compare_total(other)
            if c == -1:
                ans = other
            else:
                ans = self
            return ans._fix(context)

    def min_mag(self, other, context=None):
        """Compares the values numerically with their sign ignored."""
        other = _convert_other(other, raiseit=True)
        if context is None:
            context = getcontext()
        if self._is_special or other._is_special:
            sn = self._isnan()
            on = other._isnan()
            if sn or on:
                if on == 1:
                    pass
        if sn == 0:
            return self._fix(context)
        elif sn == 1:
            if on == 0:
                return other._fix(context)
            return self._check_nans(other, context)
        else:
            c = self.copy_abs()._cmp(other.copy_abs())
            if c == 0:
                c = self.compare_total(other)
            if c == -1:
                ans = self
            else:
                ans = other
            return ans._fix(context)

    def next_minus(self, context=None):
        """Returns the largest representable number smaller than itself."""
        if context is None:
            context = getcontext()
        ans = self._check_nans(context=context)
        if ans:
            return ans
        elif self._isinfinity() == -1:
            return _NegativeInfinity
        elif self._isinfinity() == 1:
            return _dec_from_triple(0, "9" * context.prec, context.Etop())
        context = context.copy()
        context._set_rounding(ROUND_FLOOR)
        context._ignore_all_flags()
        new_self = self._fix(context)
        if new_self != self:
            return new_self
        else:
            return self.__sub__(_dec_from_triple(0, "1", context.Etiny() - 1), context)

    def next_plus(self, context=None):
        """Returns the smallest representable number larger than itself."""
        if context is None:
            context = getcontext()
        ans = self._check_nans(context=context)
        if ans:
            return ans
        elif self._isinfinity() == 1:
            return _Infinity
        elif self._isinfinity() == -1:
            return _dec_from_triple(1, "9" * context.prec, context.Etop())
        context = context.copy()
        context._set_rounding(ROUND_CEILING)
        context._ignore_all_flags()
        new_self = self._fix(context)
        if new_self != self:
            return new_self
        else:
            return self.__add__(_dec_from_triple(0, "1", context.Etiny() - 1), context)

    def next_toward(self, other, context=None):
        """Returns the number closest to self, in the direction towards other.
        
        The result is the closest representable number to self
        (excluding self) that is in the direction towards other,
        unless both have the same value.  If the two operands are
        numerically equal, then the result is a copy of self with the
        sign set to be the same as the sign of other.
        """
        other = _convert_other(other, raiseit=True)
        if context is None:
            context = getcontext()
        ans = self._check_nans(other, context)
        if ans:
            return ans
        else:
            comparison = self._cmp(other)
            if comparison == 0:
                return self.copy_sign(other)
            if comparison == -1:
                ans = self.next_plus(context)
            else:
                ans = self.next_minus(context)
            if ans._isinfinity():
                context._raise_error(
                    Overflow, "Infinite result from next_toward", ans._sign
                )
                context._raise_error(Inexact)
                context._raise_error(Rounded)
            else:
                if ans.adjusted() < context.Emin:
                    context._raise_error(Underflow)
                    context._raise_error(Subnormal)
                    context._raise_error(Inexact)
                    context._raise_error(Rounded)
                    if not ans:
                        context._raise_error(Clamped)
            return ans

    def number_class(self, context=None):
        """Returns an indication of the class of self.
        
        The class is one of the following strings:
          sNaN
          NaN
          -Infinity
          -Normal
          -Subnormal
          -Zero
          +Zero
          +Subnormal
          +Normal
          +Infinity
        """
        if self.is_snan():
            return "sNaN"
        elif self.is_qnan():
            return "NaN"
        inf = self._isinfinity()
        if inf == 1:
            return "+Infinity"
        elif inf == -1:
            return "-Infinity"
        elif self.is_zero():
            if self._sign:
                return "-Zero"
            return "+Zero"
        if context is None:
            context = getcontext()
        if self.is_subnormal(context=context):
            if self._sign:
                return "-Subnormal"
            return "+Subnormal"
        elif self._sign:
            return "-Normal"
        else:
            return "+Normal"

    def radix(self):
        """Just returns 10, as this is Decimal, :)"""
        return Decimal(10)

    def rotate(self, other, context=None):
        """Returns a rotated copy of self, value-of-other times."""
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        ans = self._check_nans(other, context)
        if ans:
            return ans
        elif other._exp != 0:
            return context._raise_error(InvalidOperation)
        elif not -context.prec <= int(other) <= context.prec:
            return context._raise_error(InvalidOperation)
        elif self._isinfinity():
            return Decimal(self)
        else:
            torot = int(other)
            rotdig = self._int
            topad = context.prec - len(rotdig)
            if topad > 0:
                rotdig = "0" * topad + rotdig
            else:
                if topad < 0:
                    rotdig = rotdig[-topad:]
            rotated = rotdig[torot:] + rotdig[:torot]
            return _dec_from_triple(self._sign, rotated.lstrip("0") or "0", self._exp)

    def scaleb(self, other, context=None):
        """Returns self operand after adding the second value to its exp."""
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        ans = self._check_nans(other, context)
        if ans:
            return ans
        elif other._exp != 0:
            return context._raise_error(InvalidOperation)
        else:
            liminf = -2 * (context.Emax + context.prec)
            limsup = 2 * (context.Emax + context.prec)
            if not liminf <= int(other) <= limsup:
                return context._raise_error(InvalidOperation)
            elif self._isinfinity():
                return Decimal(self)
            d = _dec_from_triple(self._sign, self._int, self._exp + int(other))
            d = d._fix(context)
            return d

    def shift(self, other, context=None):
        """Returns a shifted copy of self, value-of-other times."""
        if context is None:
            context = getcontext()
        other = _convert_other(other, raiseit=True)
        ans = self._check_nans(other, context)
        if ans:
            return ans
        elif other._exp != 0:
            return context._raise_error(InvalidOperation)
        elif not -context.prec <= int(other) <= context.prec:
            return context._raise_error(InvalidOperation)
        elif self._isinfinity():
            return Decimal(self)
        else:
            torot = int(other)
            rotdig = self._int
            topad = context.prec - len(rotdig)
            if topad > 0:
                rotdig = "0" * topad + rotdig
            else:
                if topad < 0:
                    rotdig = rotdig[-topad:]
            if torot < 0:
                shifted = rotdig[:torot]
            else:
                shifted = rotdig + "0" * torot
                shifted = shifted[-context.prec :]
            return _dec_from_triple(self._sign, shifted.lstrip("0") or "0", self._exp)

    def __reduce__(self):
        return (self.__class__, (str(self),))

    def __copy__(self):
        if type(self) is Decimal:
            return self
        else:
            return self.__class__(str(self))

    def __deepcopy__(self, memo):
        if type(self) is Decimal:
            return self
        else:
            return self.__class__(str(self))

    def __format__(self, specifier, context=None, _localeconv=None):
        """Format a Decimal instance according to the given specifier.
        
        The specifier should be a standard format specifier, with the
        form described in PEP 3101.  Formatting types 'e', 'E', 'f',
        'F', 'g', 'G', 'n' and '%' are supported.  If the formatting
        type is omitted it defaults to 'g' or 'G', depending on the
        value of context.capitals.
        """
        if context is None:
            context = getcontext()
        spec = _parse_format_specifier(specifier, _localeconv=_localeconv)
        if self._is_special:
            sign = _format_sign(self._sign, spec)
            body = str(self.copy_abs())
            if spec["type"] == "%":
                body += "%"
            return _format_align(sign, body, spec)
        if spec["type"] is None:
            spec["type"] = ["g", "G"][context.capitals]
        if spec["type"] == "%":
            self = _dec_from_triple(self._sign, self._int, self._exp + 2)
        rounding = context.rounding
        precision = spec["precision"]
        if precision is not None:
            if spec["type"] in "eE":
                self = self._round(precision + 1, rounding)
            else:
                if spec["type"] in "fF%":
                    self = self._rescale(-precision, rounding)
                else:
                    if spec["type"] in "gG":
                        if len(self._int) > precision:
                            self = self._round(precision, rounding)
                if not self:
                    if self._exp > 0:
                        if spec["type"] in "fF%":
                            self = self._rescale(0, rounding)
                leftdigits = self._exp + len(self._int)
        if spec["type"] in "eE":
            if not self:
                if precision is not None:
                    dotplace = 1 - precision
                dotplace = 1
        else:
            if spec["type"] in "fF%":
                dotplace = leftdigits
            else:
                if spec["type"] in "gG":
                    if self._exp <= 0:
                        if leftdigits > -6:
                            dotplace = leftdigits
                        dotplace = 1
            if dotplace < 0:
                intpart = "0"
                fracpart = "0" * -dotplace + self._int
            else:
                if dotplace > len(self._int):
                    intpart = self._int + "0" * (dotplace - len(self._int))
                    fracpart = ""
                else:
                    intpart = self._int[:dotplace] or "0"
                    fracpart = self._int[dotplace:]
                exp = leftdigits - dotplace
                return _format_number(self._sign, intpart, fracpart, exp, spec)


def _dec_from_triple(sign, coefficient, exponent, special=False):
    """Create a decimal instance directly, without any validation,
    normalization (e.g. removal of leading zeros) or argument
    conversion.
    
    This function is for *internal use only*.
    """
    self = object.__new__(Decimal)
    self._sign = sign
    self._int = coefficient
    self._exp = exponent
    self._is_special = special
    return self


_numbers.Number.register(Decimal)


class _ContextManager(object):
    """Context manager class to support localcontext().
    
      Sets a copy of the supplied context in __enter__() and restores
      the previous decimal context in __exit__()
    """

    def __init__(self, new_context):
        self.new_context = new_context.copy()

    def __enter__(self):
        self.saved_context = getcontext()
        setcontext(self.new_context)
        return self.new_context

    def __exit__(self, t, v, tb):
        setcontext(self.saved_context)


class Context(object):
    """Contains the context for a Decimal instance.
    
    Contains:
    prec - precision (for use in rounding, division, square roots..)
    rounding - rounding type (how you round)
    traps - If traps[exception] = 1, then the exception is
                    raised when it is caused.  Otherwise, a value is
                    substituted in.
    flags  - When an exception is caused, flags[exception] is set.
             (Whether or not the trap_enabler is set)
             Should be reset by user of Decimal instance.
    Emin -   Minimum exponent
    Emax -   Maximum exponent
    capitals -      If 1, 1*10^1 is printed as 1E+1.
                    If 0, printed as 1e1
    clamp -  If 1, change exponents if too high (Default 0)
    """

    def __init__(
        self,
        prec=None,
        rounding=None,
        Emin=None,
        Emax=None,
        capitals=None,
        clamp=None,
        flags=None,
        traps=None,
        _ignored_flags=None,
    ):
        try:
            dc = DefaultContext
        except NameError:
            pass

        self.prec = prec if prec is not None else dc.prec
        self.rounding = rounding if rounding is not None else dc.rounding
        self.Emin = Emin if Emin is not None else dc.Emin
        self.Emax = Emax if Emax is not None else dc.Emax
        self.capitals = capitals if capitals is not None else dc.capitals
        self.clamp = clamp if clamp is not None else dc.clamp
        if _ignored_flags is None:
            self._ignored_flags = []
        else:
            self._ignored_flags = _ignored_flags
        if traps is None:
            self.traps = dc.traps.copy()
        else:
            if not isinstance(traps, dict):
                self.traps = dict(((s, int(s in traps)) for s in _signals + traps))
            else:
                self.traps = traps
            if flags is None:
                self.flags = dict.fromkeys(_signals, 0)
            else:
                if not isinstance(flags, dict):
                    self.flags = dict(((s, int(s in flags)) for s in _signals + flags))
                else:
                    self.flags = flags

    def _set_integer_check(self, name, value, vmin, vmax):
        if not isinstance(value, int):
            raise TypeError("%s must be an integer" % name)
        if vmin == "-inf":
            if value > vmax:
                raise ValueError(
                    "%s must be in [%s, %d]. got: %s" % (name, vmin, vmax, value)
                )
        if vmax == "inf":
            if value < vmin:
                raise ValueError(
                    "%s must be in [%d, %s]. got: %s" % (name, vmin, vmax, value)
                )
            else:
                if value < vmin or value > vmax:
                    raise ValueError(
                        "%s must be in [%d, %d]. got %s" % (name, vmin, vmax, value)
                    )
            return object.__setattr__(self, name, value)

    def _set_signal_dict(self, name, d):
        if not isinstance(d, dict):
            raise TypeError("%s must be a signal dict" % d)
        for key in d:
            if key not in _signals:
                raise KeyError("%s is not a valid signal dict" % d)

        for key in _signals:
            if key not in d:
                raise KeyError("%s is not a valid signal dict" % d)

        return object.__setattr__(self, name, d)

    def __setattr__(self, name, value):
        if name == "prec":
            return self._set_integer_check(name, value, 1, "inf")
        if name == "Emin":
            return self._set_integer_check(name, value, "-inf", 0)
        if name == "Emax":
            return self._set_integer_check(name, value, 0, "inf")
        if name == "capitals":
            return self._set_integer_check(name, value, 0, 1)
        if name == "clamp":
            return self._set_integer_check(name, value, 0, 1)
        if name == "rounding":
            if value not in _rounding_modes:
                raise TypeError("%s: invalid rounding mode" % value)
            return object.__setattr__(self, name, value)
        if name == "flags" or name == "traps":
            return self._set_signal_dict(name, value)
        if name == "_ignored_flags":
            return object.__setattr__(self, name, value)
        raise AttributeError("'decimal.Context' object has no attribute '%s'" % name)

    def __delattr__(self, name):
        raise AttributeError("%s cannot be deleted" % name)

    def __reduce__(self):
        flags = [sig for sig, v in self.flags.items() if v]
        traps = [sig for sig, v in self.traps.items() if v]
        return (
            self.__class__,
            (
                self.prec,
                self.rounding,
                self.Emin,
                self.Emax,
                self.capitals,
                self.clamp,
                flags,
                traps,
            ),
        )

    def __repr__(self):
        """Show the current context."""
        s = []
        s.append(
            "Context(prec=%(prec)d, rounding=%(rounding)s, Emin=%(Emin)d, Emax=%(Emax)d, capitals=%(capitals)d, clamp=%(clamp)d"
            % vars(self)
        )
        names = [f.__name__ for f, v in self.flags.items() if v]
        s.append("flags=[" + (", ").join(names) + "]")
        names = [t.__name__ for t, v in self.traps.items() if v]
        s.append("traps=[" + (", ").join(names) + "]")
        return (", ").join(s) + ")"

    def clear_flags(self):
        """Reset all flags to zero"""
        for flag in self.flags:
            self.flags[flag] = 0

    def clear_traps(self):
        """Reset all traps to zero"""
        for flag in self.traps:
            self.traps[flag] = 0

    def _shallow_copy(self):
        """Returns a shallow copy from self."""
        nc = Context(
            self.prec,
            self.rounding,
            self.Emin,
            self.Emax,
            self.capitals,
            self.clamp,
            self.flags,
            self.traps,
            self._ignored_flags,
        )
        return nc

    def copy(self):
        """Returns a deep copy from self."""
        nc = Context(
            self.prec,
            self.rounding,
            self.Emin,
            self.Emax,
            self.capitals,
            self.clamp,
            self.flags.copy(),
            self.traps.copy(),
            self._ignored_flags,
        )
        return nc

    __copy__ = copy

    def _raise_error(self, condition, explanation=None, *args):
        """Handles an error
        
        If the flag is in _ignored_flags, returns the default response.
        Otherwise, it sets the flag, then, if the corresponding
        trap_enabler is set, it reraises the exception.  Otherwise, it returns
        the default value after setting the flag.
        """
        error = _condition_map.get(condition, condition)
        if error in self._ignored_flags:
            return error().handle(self, *args)
        self.flags[error] = 1
        if not self.traps[error]:
            return condition().handle(self, *args)
        raise error(explanation)

    def _ignore_all_flags(self):
        """Ignore all flags, if they are raised"""
        return self._ignore_flags(*_signals)

    def _ignore_flags(self, *flags):
        """Ignore the flags, if they are raised"""
        self._ignored_flags = self._ignored_flags + list(flags)
        return list(flags)

    def _regard_flags(self, *flags):
        """Stop ignoring the flags, if they are raised"""
        if flags:
            if isinstance(flags[0], (tuple, list)):
                flags = flags[0]
        for flag in flags:
            self._ignored_flags.remove(flag)

    __hash__ = None

    def Etiny(self):
        """Returns Etiny (= Emin - prec + 1)"""
        return int(self.Emin - self.prec + 1)

    def Etop(self):
        """Returns maximum exponent (= Emax - prec + 1)"""
        return int(self.Emax - self.prec + 1)

    def _set_rounding(self, type):
        """Sets the rounding type.
        
        Sets the rounding type, and returns the current (previous)
        rounding type.  Often used like:
        
        context = context.copy()
        # so you don't change the calling context
        # if an error occurs in the middle.
        rounding = context._set_rounding(ROUND_UP)
        val = self.__sub__(other, context=context)
        context._set_rounding(rounding)
        
        This will make it round up for that operation.
        """
        rounding = self.rounding
        self.rounding = type
        return rounding

    def create_decimal(self, num="0"):
        """Creates a new Decimal instance but using self as context.
        
        This method implements the to-number operation of the
        IBM Decimal specification."""
        if isinstance(num, str):
            if num != num.strip() or "_" in num:
                return self._raise_error(
                    ConversionSyntax,
                    "trailing or leading whitespace and underscores are not permitted.",
                )
            d = Decimal(num, context=self)
            if d._isnan():
                if len(d._int) > self.prec - self.clamp:
                    return self._raise_error(
                        ConversionSyntax, "diagnostic info too long in NaN"
                    )
                return d._fix(self)

    def create_decimal_from_float(self, f):
        """Creates a new Decimal instance from a float but rounding using self
        as the context.
        
        >>> context = Context(prec=5, rounding=ROUND_DOWN)
        >>> context.create_decimal_from_float(3.1415926535897932)
        Decimal('3.1415')
        >>> context = Context(prec=5, traps=[Inexact])
        >>> context.create_decimal_from_float(3.1415926535897932)
        Traceback (most recent call last):
            ...
        decimal.Inexact: None
        
        """
        d = Decimal.from_float(f)
        return d._fix(self)

    def abs(self, a):
        """Returns the absolute value of the operand.
        
        If the operand is negative, the result is the same as using the minus
        operation on the operand.  Otherwise, the result is the same as using
        the plus operation on the operand.
        
        >>> ExtendedContext.abs(Decimal('2.1'))
        Decimal('2.1')
        >>> ExtendedContext.abs(Decimal('-100'))
        Decimal('100')
        >>> ExtendedContext.abs(Decimal('101.5'))
        Decimal('101.5')
        >>> ExtendedContext.abs(Decimal('-101.5'))
        Decimal('101.5')
        >>> ExtendedContext.abs(-1)
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        return a.__abs__(context=self)

    def add(self, a, b):
        """Return the sum of the two operands.
        
        >>> ExtendedContext.add(Decimal('12'), Decimal('7.00'))
        Decimal('19.00')
        >>> ExtendedContext.add(Decimal('1E+2'), Decimal('1.01E+4'))
        Decimal('1.02E+4')
        >>> ExtendedContext.add(1, Decimal(2))
        Decimal('3')
        >>> ExtendedContext.add(Decimal(8), 5)
        Decimal('13')
        >>> ExtendedContext.add(5, 5)
        Decimal('10')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__add__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def _apply(self, a):
        return str(a._fix(self))

    def canonical(self, a):
        """Returns the same Decimal object.
        
        As we do not have different encodings for the same number, the
        received object already is in its canonical form.
        
        >>> ExtendedContext.canonical(Decimal('2.50'))
        Decimal('2.50')
        """
        if not isinstance(a, Decimal):
            raise TypeError("canonical requires a Decimal as an argument.")
        return a.canonical()

    def compare(self, a, b):
        """Compares values numerically.
        
        If the signs of the operands differ, a value representing each operand
        ('-1' if the operand is less than zero, '0' if the operand is zero or
        negative zero, or '1' if the operand is greater than zero) is used in
        place of that operand for the comparison instead of the actual
        operand.
        
        The comparison is then effected by subtracting the second operand from
        the first and then returning a value according to the result of the
        subtraction: '-1' if the result is less than zero, '0' if the result is
        zero or negative zero, or '1' if the result is greater than zero.
        
        >>> ExtendedContext.compare(Decimal('2.1'), Decimal('3'))
        Decimal('-1')
        >>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.1'))
        Decimal('0')
        >>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.10'))
        Decimal('0')
        >>> ExtendedContext.compare(Decimal('3'), Decimal('2.1'))
        Decimal('1')
        >>> ExtendedContext.compare(Decimal('2.1'), Decimal('-3'))
        Decimal('1')
        >>> ExtendedContext.compare(Decimal('-3'), Decimal('2.1'))
        Decimal('-1')
        >>> ExtendedContext.compare(1, 2)
        Decimal('-1')
        >>> ExtendedContext.compare(Decimal(1), 2)
        Decimal('-1')
        >>> ExtendedContext.compare(1, Decimal(2))
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.compare(b, context=self)

    def compare_signal(self, a, b):
        """Compares the values of the two operands numerically.
        
        It's pretty much like compare(), but all NaNs signal, with signaling
        NaNs taking precedence over quiet NaNs.
        
        >>> c = ExtendedContext
        >>> c.compare_signal(Decimal('2.1'), Decimal('3'))
        Decimal('-1')
        >>> c.compare_signal(Decimal('2.1'), Decimal('2.1'))
        Decimal('0')
        >>> c.flags[InvalidOperation] = 0
        >>> print(c.flags[InvalidOperation])
        0
        >>> c.compare_signal(Decimal('NaN'), Decimal('2.1'))
        Decimal('NaN')
        >>> print(c.flags[InvalidOperation])
        1
        >>> c.flags[InvalidOperation] = 0
        >>> print(c.flags[InvalidOperation])
        0
        >>> c.compare_signal(Decimal('sNaN'), Decimal('2.1'))
        Decimal('NaN')
        >>> print(c.flags[InvalidOperation])
        1
        >>> c.compare_signal(-1, 2)
        Decimal('-1')
        >>> c.compare_signal(Decimal(-1), 2)
        Decimal('-1')
        >>> c.compare_signal(-1, Decimal(2))
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.compare_signal(b, context=self)

    def compare_total(self, a, b):
        """Compares two operands using their abstract representation.
        
        This is not like the standard compare, which use their numerical
        value. Note that a total ordering is defined for all possible abstract
        representations.
        
        >>> ExtendedContext.compare_total(Decimal('12.73'), Decimal('127.9'))
        Decimal('-1')
        >>> ExtendedContext.compare_total(Decimal('-127'),  Decimal('12'))
        Decimal('-1')
        >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.3'))
        Decimal('-1')
        >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.30'))
        Decimal('0')
        >>> ExtendedContext.compare_total(Decimal('12.3'),  Decimal('12.300'))
        Decimal('1')
        >>> ExtendedContext.compare_total(Decimal('12.3'),  Decimal('NaN'))
        Decimal('-1')
        >>> ExtendedContext.compare_total(1, 2)
        Decimal('-1')
        >>> ExtendedContext.compare_total(Decimal(1), 2)
        Decimal('-1')
        >>> ExtendedContext.compare_total(1, Decimal(2))
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.compare_total(b)

    def compare_total_mag(self, a, b):
        """Compares two operands using their abstract representation ignoring sign.
        
        Like compare_total, but with operand's sign ignored and assumed to be 0.
        """
        a = _convert_other(a, raiseit=True)
        return a.compare_total_mag(b)

    def copy_abs(self, a):
        """Returns a copy of the operand with the sign set to 0.
        
        >>> ExtendedContext.copy_abs(Decimal('2.1'))
        Decimal('2.1')
        >>> ExtendedContext.copy_abs(Decimal('-100'))
        Decimal('100')
        >>> ExtendedContext.copy_abs(-1)
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        return a.copy_abs()

    def copy_decimal(self, a):
        """Returns a copy of the decimal object.
        
        >>> ExtendedContext.copy_decimal(Decimal('2.1'))
        Decimal('2.1')
        >>> ExtendedContext.copy_decimal(Decimal('-1.00'))
        Decimal('-1.00')
        >>> ExtendedContext.copy_decimal(1)
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        return Decimal(a)

    def copy_negate(self, a):
        """Returns a copy of the operand with the sign inverted.
        
        >>> ExtendedContext.copy_negate(Decimal('101.5'))
        Decimal('-101.5')
        >>> ExtendedContext.copy_negate(Decimal('-101.5'))
        Decimal('101.5')
        >>> ExtendedContext.copy_negate(1)
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.copy_negate()

    def copy_sign(self, a, b):
        """Copies the second operand's sign to the first one.
        
        In detail, it returns a copy of the first operand with the sign
        equal to the sign of the second operand.
        
        >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('7.33'))
        Decimal('1.50')
        >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('7.33'))
        Decimal('1.50')
        >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('-7.33'))
        Decimal('-1.50')
        >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('-7.33'))
        Decimal('-1.50')
        >>> ExtendedContext.copy_sign(1, -2)
        Decimal('-1')
        >>> ExtendedContext.copy_sign(Decimal(1), -2)
        Decimal('-1')
        >>> ExtendedContext.copy_sign(1, Decimal(-2))
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.copy_sign(b)

    def divide(self, a, b):
        """Decimal division in a specified context.
        
        >>> ExtendedContext.divide(Decimal('1'), Decimal('3'))
        Decimal('0.333333333')
        >>> ExtendedContext.divide(Decimal('2'), Decimal('3'))
        Decimal('0.666666667')
        >>> ExtendedContext.divide(Decimal('5'), Decimal('2'))
        Decimal('2.5')
        >>> ExtendedContext.divide(Decimal('1'), Decimal('10'))
        Decimal('0.1')
        >>> ExtendedContext.divide(Decimal('12'), Decimal('12'))
        Decimal('1')
        >>> ExtendedContext.divide(Decimal('8.00'), Decimal('2'))
        Decimal('4.00')
        >>> ExtendedContext.divide(Decimal('2.400'), Decimal('2.0'))
        Decimal('1.20')
        >>> ExtendedContext.divide(Decimal('1000'), Decimal('100'))
        Decimal('10')
        >>> ExtendedContext.divide(Decimal('1000'), Decimal('1'))
        Decimal('1000')
        >>> ExtendedContext.divide(Decimal('2.40E+6'), Decimal('2'))
        Decimal('1.20E+6')
        >>> ExtendedContext.divide(5, 5)
        Decimal('1')
        >>> ExtendedContext.divide(Decimal(5), 5)
        Decimal('1')
        >>> ExtendedContext.divide(5, Decimal(5))
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__truediv__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def divide_int(self, a, b):
        """Divides two numbers and returns the integer part of the result.
        
        >>> ExtendedContext.divide_int(Decimal('2'), Decimal('3'))
        Decimal('0')
        >>> ExtendedContext.divide_int(Decimal('10'), Decimal('3'))
        Decimal('3')
        >>> ExtendedContext.divide_int(Decimal('1'), Decimal('0.3'))
        Decimal('3')
        >>> ExtendedContext.divide_int(10, 3)
        Decimal('3')
        >>> ExtendedContext.divide_int(Decimal(10), 3)
        Decimal('3')
        >>> ExtendedContext.divide_int(10, Decimal(3))
        Decimal('3')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__floordiv__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def divmod(self, a, b):
        """Return (a // b, a % b).
        
        >>> ExtendedContext.divmod(Decimal(8), Decimal(3))
        (Decimal('2'), Decimal('2'))
        >>> ExtendedContext.divmod(Decimal(8), Decimal(4))
        (Decimal('2'), Decimal('0'))
        >>> ExtendedContext.divmod(8, 4)
        (Decimal('2'), Decimal('0'))
        >>> ExtendedContext.divmod(Decimal(8), 4)
        (Decimal('2'), Decimal('0'))
        >>> ExtendedContext.divmod(8, Decimal(4))
        (Decimal('2'), Decimal('0'))
        """
        a = _convert_other(a, raiseit=True)
        r = a.__divmod__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def exp(self, a):
        """Returns e ** a.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.exp(Decimal('-Infinity'))
        Decimal('0')
        >>> c.exp(Decimal('-1'))
        Decimal('0.367879441')
        >>> c.exp(Decimal('0'))
        Decimal('1')
        >>> c.exp(Decimal('1'))
        Decimal('2.71828183')
        >>> c.exp(Decimal('0.693147181'))
        Decimal('2.00000000')
        >>> c.exp(Decimal('+Infinity'))
        Decimal('Infinity')
        >>> c.exp(10)
        Decimal('22026.4658')
        """
        a = _convert_other(a, raiseit=True)
        return a.exp(context=self)

    def fma(self, a, b, c):
        """Returns a multiplied by b, plus c.
        
        The first two operands are multiplied together, using multiply,
        the third operand is then added to the result of that
        multiplication, using add, all with only one final rounding.
        
        >>> ExtendedContext.fma(Decimal('3'), Decimal('5'), Decimal('7'))
        Decimal('22')
        >>> ExtendedContext.fma(Decimal('3'), Decimal('-5'), Decimal('7'))
        Decimal('-8')
        >>> ExtendedContext.fma(Decimal('888565290'), Decimal('1557.96930'), Decimal('-86087.7578'))
        Decimal('1.38435736E+12')
        >>> ExtendedContext.fma(1, 3, 4)
        Decimal('7')
        >>> ExtendedContext.fma(1, Decimal(3), 4)
        Decimal('7')
        >>> ExtendedContext.fma(1, 3, Decimal(4))
        Decimal('7')
        """
        a = _convert_other(a, raiseit=True)
        return a.fma(b, c, context=self)

    def is_canonical(self, a):
        """Return True if the operand is canonical; otherwise return False.
        
        Currently, the encoding of a Decimal instance is always
        canonical, so this method returns True for any Decimal.
        
        >>> ExtendedContext.is_canonical(Decimal('2.50'))
        True
        """
        if not isinstance(a, Decimal):
            raise TypeError("is_canonical requires a Decimal as an argument.")
        return a.is_canonical()

    def is_finite(self, a):
        """Return True if the operand is finite; otherwise return False.
        
        A Decimal instance is considered finite if it is neither
        infinite nor a NaN.
        
        >>> ExtendedContext.is_finite(Decimal('2.50'))
        True
        >>> ExtendedContext.is_finite(Decimal('-0.3'))
        True
        >>> ExtendedContext.is_finite(Decimal('0'))
        True
        >>> ExtendedContext.is_finite(Decimal('Inf'))
        False
        >>> ExtendedContext.is_finite(Decimal('NaN'))
        False
        >>> ExtendedContext.is_finite(1)
        True
        """
        a = _convert_other(a, raiseit=True)
        return a.is_finite()

    def is_infinite(self, a):
        """Return True if the operand is infinite; otherwise return False.
        
        >>> ExtendedContext.is_infinite(Decimal('2.50'))
        False
        >>> ExtendedContext.is_infinite(Decimal('-Inf'))
        True
        >>> ExtendedContext.is_infinite(Decimal('NaN'))
        False
        >>> ExtendedContext.is_infinite(1)
        False
        """
        a = _convert_other(a, raiseit=True)
        return a.is_infinite()

    def is_nan(self, a):
        """Return True if the operand is a qNaN or sNaN;
        otherwise return False.
        
        >>> ExtendedContext.is_nan(Decimal('2.50'))
        False
        >>> ExtendedContext.is_nan(Decimal('NaN'))
        True
        >>> ExtendedContext.is_nan(Decimal('-sNaN'))
        True
        >>> ExtendedContext.is_nan(1)
        False
        """
        a = _convert_other(a, raiseit=True)
        return a.is_nan()

    def is_normal(self, a):
        """Return True if the operand is a normal number;
        otherwise return False.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.is_normal(Decimal('2.50'))
        True
        >>> c.is_normal(Decimal('0.1E-999'))
        False
        >>> c.is_normal(Decimal('0.00'))
        False
        >>> c.is_normal(Decimal('-Inf'))
        False
        >>> c.is_normal(Decimal('NaN'))
        False
        >>> c.is_normal(1)
        True
        """
        a = _convert_other(a, raiseit=True)
        return a.is_normal(context=self)

    def is_qnan(self, a):
        """Return True if the operand is a quiet NaN; otherwise return False.
        
        >>> ExtendedContext.is_qnan(Decimal('2.50'))
        False
        >>> ExtendedContext.is_qnan(Decimal('NaN'))
        True
        >>> ExtendedContext.is_qnan(Decimal('sNaN'))
        False
        >>> ExtendedContext.is_qnan(1)
        False
        """
        a = _convert_other(a, raiseit=True)
        return a.is_qnan()

    def is_signed(self, a):
        """Return True if the operand is negative; otherwise return False.
        
        >>> ExtendedContext.is_signed(Decimal('2.50'))
        False
        >>> ExtendedContext.is_signed(Decimal('-12'))
        True
        >>> ExtendedContext.is_signed(Decimal('-0'))
        True
        >>> ExtendedContext.is_signed(8)
        False
        >>> ExtendedContext.is_signed(-8)
        True
        """
        a = _convert_other(a, raiseit=True)
        return a.is_signed()

    def is_snan(self, a):
        """Return True if the operand is a signaling NaN;
        otherwise return False.
        
        >>> ExtendedContext.is_snan(Decimal('2.50'))
        False
        >>> ExtendedContext.is_snan(Decimal('NaN'))
        False
        >>> ExtendedContext.is_snan(Decimal('sNaN'))
        True
        >>> ExtendedContext.is_snan(1)
        False
        """
        a = _convert_other(a, raiseit=True)
        return a.is_snan()

    def is_subnormal(self, a):
        """Return True if the operand is subnormal; otherwise return False.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.is_subnormal(Decimal('2.50'))
        False
        >>> c.is_subnormal(Decimal('0.1E-999'))
        True
        >>> c.is_subnormal(Decimal('0.00'))
        False
        >>> c.is_subnormal(Decimal('-Inf'))
        False
        >>> c.is_subnormal(Decimal('NaN'))
        False
        >>> c.is_subnormal(1)
        False
        """
        a = _convert_other(a, raiseit=True)
        return a.is_subnormal(context=self)

    def is_zero(self, a):
        """Return True if the operand is a zero; otherwise return False.
        
        >>> ExtendedContext.is_zero(Decimal('0'))
        True
        >>> ExtendedContext.is_zero(Decimal('2.50'))
        False
        >>> ExtendedContext.is_zero(Decimal('-0E+2'))
        True
        >>> ExtendedContext.is_zero(1)
        False
        >>> ExtendedContext.is_zero(0)
        True
        """
        a = _convert_other(a, raiseit=True)
        return a.is_zero()

    def ln(self, a):
        """Returns the natural (base e) logarithm of the operand.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.ln(Decimal('0'))
        Decimal('-Infinity')
        >>> c.ln(Decimal('1.000'))
        Decimal('0')
        >>> c.ln(Decimal('2.71828183'))
        Decimal('1.00000000')
        >>> c.ln(Decimal('10'))
        Decimal('2.30258509')
        >>> c.ln(Decimal('+Infinity'))
        Decimal('Infinity')
        >>> c.ln(1)
        Decimal('0')
        """
        a = _convert_other(a, raiseit=True)
        return a.ln(context=self)

    def log10(self, a):
        """Returns the base 10 logarithm of the operand.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.log10(Decimal('0'))
        Decimal('-Infinity')
        >>> c.log10(Decimal('0.001'))
        Decimal('-3')
        >>> c.log10(Decimal('1.000'))
        Decimal('0')
        >>> c.log10(Decimal('2'))
        Decimal('0.301029996')
        >>> c.log10(Decimal('10'))
        Decimal('1')
        >>> c.log10(Decimal('70'))
        Decimal('1.84509804')
        >>> c.log10(Decimal('+Infinity'))
        Decimal('Infinity')
        >>> c.log10(0)
        Decimal('-Infinity')
        >>> c.log10(1)
        Decimal('0')
        """
        a = _convert_other(a, raiseit=True)
        return a.log10(context=self)

    def logb(self, a):
        """ Returns the exponent of the magnitude of the operand's MSD.
        
        The result is the integer which is the exponent of the magnitude
        of the most significant digit of the operand (as though the
        operand were truncated to a single digit while maintaining the
        value of that digit and without limiting the resulting exponent).
        
        >>> ExtendedContext.logb(Decimal('250'))
        Decimal('2')
        >>> ExtendedContext.logb(Decimal('2.50'))
        Decimal('0')
        >>> ExtendedContext.logb(Decimal('0.03'))
        Decimal('-2')
        >>> ExtendedContext.logb(Decimal('0'))
        Decimal('-Infinity')
        >>> ExtendedContext.logb(1)
        Decimal('0')
        >>> ExtendedContext.logb(10)
        Decimal('1')
        >>> ExtendedContext.logb(100)
        Decimal('2')
        """
        a = _convert_other(a, raiseit=True)
        return a.logb(context=self)

    def logical_and(self, a, b):
        """Applies the logical operation 'and' between each operand's digits.
        
        The operands must be both logical numbers.
        
        >>> ExtendedContext.logical_and(Decimal('0'), Decimal('0'))
        Decimal('0')
        >>> ExtendedContext.logical_and(Decimal('0'), Decimal('1'))
        Decimal('0')
        >>> ExtendedContext.logical_and(Decimal('1'), Decimal('0'))
        Decimal('0')
        >>> ExtendedContext.logical_and(Decimal('1'), Decimal('1'))
        Decimal('1')
        >>> ExtendedContext.logical_and(Decimal('1100'), Decimal('1010'))
        Decimal('1000')
        >>> ExtendedContext.logical_and(Decimal('1111'), Decimal('10'))
        Decimal('10')
        >>> ExtendedContext.logical_and(110, 1101)
        Decimal('100')
        >>> ExtendedContext.logical_and(Decimal(110), 1101)
        Decimal('100')
        >>> ExtendedContext.logical_and(110, Decimal(1101))
        Decimal('100')
        """
        a = _convert_other(a, raiseit=True)
        return a.logical_and(b, context=self)

    def logical_invert(self, a):
        """Invert all the digits in the operand.
        
        The operand must be a logical number.
        
        >>> ExtendedContext.logical_invert(Decimal('0'))
        Decimal('111111111')
        >>> ExtendedContext.logical_invert(Decimal('1'))
        Decimal('111111110')
        >>> ExtendedContext.logical_invert(Decimal('111111111'))
        Decimal('0')
        >>> ExtendedContext.logical_invert(Decimal('101010101'))
        Decimal('10101010')
        >>> ExtendedContext.logical_invert(1101)
        Decimal('111110010')
        """
        a = _convert_other(a, raiseit=True)
        return a.logical_invert(context=self)

    def logical_or(self, a, b):
        """Applies the logical operation 'or' between each operand's digits.
        
        The operands must be both logical numbers.
        
        >>> ExtendedContext.logical_or(Decimal('0'), Decimal('0'))
        Decimal('0')
        >>> ExtendedContext.logical_or(Decimal('0'), Decimal('1'))
        Decimal('1')
        >>> ExtendedContext.logical_or(Decimal('1'), Decimal('0'))
        Decimal('1')
        >>> ExtendedContext.logical_or(Decimal('1'), Decimal('1'))
        Decimal('1')
        >>> ExtendedContext.logical_or(Decimal('1100'), Decimal('1010'))
        Decimal('1110')
        >>> ExtendedContext.logical_or(Decimal('1110'), Decimal('10'))
        Decimal('1110')
        >>> ExtendedContext.logical_or(110, 1101)
        Decimal('1111')
        >>> ExtendedContext.logical_or(Decimal(110), 1101)
        Decimal('1111')
        >>> ExtendedContext.logical_or(110, Decimal(1101))
        Decimal('1111')
        """
        a = _convert_other(a, raiseit=True)
        return a.logical_or(b, context=self)

    def logical_xor(self, a, b):
        """Applies the logical operation 'xor' between each operand's digits.
        
        The operands must be both logical numbers.
        
        >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('0'))
        Decimal('0')
        >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('1'))
        Decimal('1')
        >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('0'))
        Decimal('1')
        >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('1'))
        Decimal('0')
        >>> ExtendedContext.logical_xor(Decimal('1100'), Decimal('1010'))
        Decimal('110')
        >>> ExtendedContext.logical_xor(Decimal('1111'), Decimal('10'))
        Decimal('1101')
        >>> ExtendedContext.logical_xor(110, 1101)
        Decimal('1011')
        >>> ExtendedContext.logical_xor(Decimal(110), 1101)
        Decimal('1011')
        >>> ExtendedContext.logical_xor(110, Decimal(1101))
        Decimal('1011')
        """
        a = _convert_other(a, raiseit=True)
        return a.logical_xor(b, context=self)

    def max(self, a, b):
        """max compares two values numerically and returns the maximum.
        
        If either operand is a NaN then the general rules apply.
        Otherwise, the operands are compared as though by the compare
        operation.  If they are numerically equal then the left-hand operand
        is chosen as the result.  Otherwise the maximum (closer to positive
        infinity) of the two operands is chosen as the result.
        
        >>> ExtendedContext.max(Decimal('3'), Decimal('2'))
        Decimal('3')
        >>> ExtendedContext.max(Decimal('-10'), Decimal('3'))
        Decimal('3')
        >>> ExtendedContext.max(Decimal('1.0'), Decimal('1'))
        Decimal('1')
        >>> ExtendedContext.max(Decimal('7'), Decimal('NaN'))
        Decimal('7')
        >>> ExtendedContext.max(1, 2)
        Decimal('2')
        >>> ExtendedContext.max(Decimal(1), 2)
        Decimal('2')
        >>> ExtendedContext.max(1, Decimal(2))
        Decimal('2')
        """
        a = _convert_other(a, raiseit=True)
        return a.max(b, context=self)

    def max_mag(self, a, b):
        """Compares the values numerically with their sign ignored.
        
        >>> ExtendedContext.max_mag(Decimal('7'), Decimal('NaN'))
        Decimal('7')
        >>> ExtendedContext.max_mag(Decimal('7'), Decimal('-10'))
        Decimal('-10')
        >>> ExtendedContext.max_mag(1, -2)
        Decimal('-2')
        >>> ExtendedContext.max_mag(Decimal(1), -2)
        Decimal('-2')
        >>> ExtendedContext.max_mag(1, Decimal(-2))
        Decimal('-2')
        """
        a = _convert_other(a, raiseit=True)
        return a.max_mag(b, context=self)

    def min(self, a, b):
        """min compares two values numerically and returns the minimum.
        
        If either operand is a NaN then the general rules apply.
        Otherwise, the operands are compared as though by the compare
        operation.  If they are numerically equal then the left-hand operand
        is chosen as the result.  Otherwise the minimum (closer to negative
        infinity) of the two operands is chosen as the result.
        
        >>> ExtendedContext.min(Decimal('3'), Decimal('2'))
        Decimal('2')
        >>> ExtendedContext.min(Decimal('-10'), Decimal('3'))
        Decimal('-10')
        >>> ExtendedContext.min(Decimal('1.0'), Decimal('1'))
        Decimal('1.0')
        >>> ExtendedContext.min(Decimal('7'), Decimal('NaN'))
        Decimal('7')
        >>> ExtendedContext.min(1, 2)
        Decimal('1')
        >>> ExtendedContext.min(Decimal(1), 2)
        Decimal('1')
        >>> ExtendedContext.min(1, Decimal(29))
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        return a.min(b, context=self)

    def min_mag(self, a, b):
        """Compares the values numerically with their sign ignored.
        
        >>> ExtendedContext.min_mag(Decimal('3'), Decimal('-2'))
        Decimal('-2')
        >>> ExtendedContext.min_mag(Decimal('-3'), Decimal('NaN'))
        Decimal('-3')
        >>> ExtendedContext.min_mag(1, -2)
        Decimal('1')
        >>> ExtendedContext.min_mag(Decimal(1), -2)
        Decimal('1')
        >>> ExtendedContext.min_mag(1, Decimal(-2))
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        return a.min_mag(b, context=self)

    def minus(self, a):
        """Minus corresponds to unary prefix minus in Python.
        
        The operation is evaluated using the same rules as subtract; the
        operation minus(a) is calculated as subtract('0', a) where the '0'
        has the same exponent as the operand.
        
        >>> ExtendedContext.minus(Decimal('1.3'))
        Decimal('-1.3')
        >>> ExtendedContext.minus(Decimal('-1.3'))
        Decimal('1.3')
        >>> ExtendedContext.minus(1)
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.__neg__(context=self)

    def multiply(self, a, b):
        """multiply multiplies two operands.
        
        If either operand is a special value then the general rules apply.
        Otherwise, the operands are multiplied together
        ('long multiplication'), resulting in a number which may be as long as
        the sum of the lengths of the two operands.
        
        >>> ExtendedContext.multiply(Decimal('1.20'), Decimal('3'))
        Decimal('3.60')
        >>> ExtendedContext.multiply(Decimal('7'), Decimal('3'))
        Decimal('21')
        >>> ExtendedContext.multiply(Decimal('0.9'), Decimal('0.8'))
        Decimal('0.72')
        >>> ExtendedContext.multiply(Decimal('0.9'), Decimal('-0'))
        Decimal('-0.0')
        >>> ExtendedContext.multiply(Decimal('654321'), Decimal('654321'))
        Decimal('4.28135971E+11')
        >>> ExtendedContext.multiply(7, 7)
        Decimal('49')
        >>> ExtendedContext.multiply(Decimal(7), 7)
        Decimal('49')
        >>> ExtendedContext.multiply(7, Decimal(7))
        Decimal('49')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__mul__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def next_minus(self, a):
        """Returns the largest representable number smaller than a.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> ExtendedContext.next_minus(Decimal('1'))
        Decimal('0.999999999')
        >>> c.next_minus(Decimal('1E-1007'))
        Decimal('0E-1007')
        >>> ExtendedContext.next_minus(Decimal('-1.00000003'))
        Decimal('-1.00000004')
        >>> c.next_minus(Decimal('Infinity'))
        Decimal('9.99999999E+999')
        >>> c.next_minus(1)
        Decimal('0.999999999')
        """
        a = _convert_other(a, raiseit=True)
        return a.next_minus(context=self)

    def next_plus(self, a):
        """Returns the smallest representable number larger than a.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> ExtendedContext.next_plus(Decimal('1'))
        Decimal('1.00000001')
        >>> c.next_plus(Decimal('-1E-1007'))
        Decimal('-0E-1007')
        >>> ExtendedContext.next_plus(Decimal('-1.00000003'))
        Decimal('-1.00000002')
        >>> c.next_plus(Decimal('-Infinity'))
        Decimal('-9.99999999E+999')
        >>> c.next_plus(1)
        Decimal('1.00000001')
        """
        a = _convert_other(a, raiseit=True)
        return a.next_plus(context=self)

    def next_toward(self, a, b):
        """Returns the number closest to a, in direction towards b.
        
        The result is the closest representable number from the first
        operand (but not the first operand) that is in the direction
        towards the second operand, unless the operands have the same
        value.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.next_toward(Decimal('1'), Decimal('2'))
        Decimal('1.00000001')
        >>> c.next_toward(Decimal('-1E-1007'), Decimal('1'))
        Decimal('-0E-1007')
        >>> c.next_toward(Decimal('-1.00000003'), Decimal('0'))
        Decimal('-1.00000002')
        >>> c.next_toward(Decimal('1'), Decimal('0'))
        Decimal('0.999999999')
        >>> c.next_toward(Decimal('1E-1007'), Decimal('-100'))
        Decimal('0E-1007')
        >>> c.next_toward(Decimal('-1.00000003'), Decimal('-10'))
        Decimal('-1.00000004')
        >>> c.next_toward(Decimal('0.00'), Decimal('-0.0000'))
        Decimal('-0.00')
        >>> c.next_toward(0, 1)
        Decimal('1E-1007')
        >>> c.next_toward(Decimal(0), 1)
        Decimal('1E-1007')
        >>> c.next_toward(0, Decimal(1))
        Decimal('1E-1007')
        """
        a = _convert_other(a, raiseit=True)
        return a.next_toward(b, context=self)

    def normalize(self, a):
        """normalize reduces an operand to its simplest form.
        
        Essentially a plus operation with all trailing zeros removed from the
        result.
        
        >>> ExtendedContext.normalize(Decimal('2.1'))
        Decimal('2.1')
        >>> ExtendedContext.normalize(Decimal('-2.0'))
        Decimal('-2')
        >>> ExtendedContext.normalize(Decimal('1.200'))
        Decimal('1.2')
        >>> ExtendedContext.normalize(Decimal('-120'))
        Decimal('-1.2E+2')
        >>> ExtendedContext.normalize(Decimal('120.00'))
        Decimal('1.2E+2')
        >>> ExtendedContext.normalize(Decimal('0.00'))
        Decimal('0')
        >>> ExtendedContext.normalize(6)
        Decimal('6')
        """
        a = _convert_other(a, raiseit=True)
        return a.normalize(context=self)

    def number_class(self, a):
        """Returns an indication of the class of the operand.
        
        The class is one of the following strings:
          -sNaN
          -NaN
          -Infinity
          -Normal
          -Subnormal
          -Zero
          +Zero
          +Subnormal
          +Normal
          +Infinity
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.number_class(Decimal('Infinity'))
        '+Infinity'
        >>> c.number_class(Decimal('1E-10'))
        '+Normal'
        >>> c.number_class(Decimal('2.50'))
        '+Normal'
        >>> c.number_class(Decimal('0.1E-999'))
        '+Subnormal'
        >>> c.number_class(Decimal('0'))
        '+Zero'
        >>> c.number_class(Decimal('-0'))
        '-Zero'
        >>> c.number_class(Decimal('-0.1E-999'))
        '-Subnormal'
        >>> c.number_class(Decimal('-1E-10'))
        '-Normal'
        >>> c.number_class(Decimal('-2.50'))
        '-Normal'
        >>> c.number_class(Decimal('-Infinity'))
        '-Infinity'
        >>> c.number_class(Decimal('NaN'))
        'NaN'
        >>> c.number_class(Decimal('-NaN'))
        'NaN'
        >>> c.number_class(Decimal('sNaN'))
        'sNaN'
        >>> c.number_class(123)
        '+Normal'
        """
        a = _convert_other(a, raiseit=True)
        return a.number_class(context=self)

    def plus(self, a):
        """Plus corresponds to unary prefix plus in Python.
        
        The operation is evaluated using the same rules as add; the
        operation plus(a) is calculated as add('0', a) where the '0'
        has the same exponent as the operand.
        
        >>> ExtendedContext.plus(Decimal('1.3'))
        Decimal('1.3')
        >>> ExtendedContext.plus(Decimal('-1.3'))
        Decimal('-1.3')
        >>> ExtendedContext.plus(-1)
        Decimal('-1')
        """
        a = _convert_other(a, raiseit=True)
        return a.__pos__(context=self)

    def power(self, a, b, modulo=None):
        """Raises a to the power of b, to modulo if given.
        
        With two arguments, compute a**b.  If a is negative then b
        must be integral.  The result will be inexact unless b is
        integral and the result is finite and can be expressed exactly
        in 'precision' digits.
        
        With three arguments, compute (a**b) % modulo.  For the
        three argument form, the following restrictions on the
        arguments hold:
        
         - all three arguments must be integral
         - b must be nonnegative
         - at least one of a or b must be nonzero
         - modulo must be nonzero and have at most 'precision' digits
        
        The result of pow(a, b, modulo) is identical to the result
        that would be obtained by computing (a**b) % modulo with
        unbounded precision, but is computed more efficiently.  It is
        always exact.
        
        >>> c = ExtendedContext.copy()
        >>> c.Emin = -999
        >>> c.Emax = 999
        >>> c.power(Decimal('2'), Decimal('3'))
        Decimal('8')
        >>> c.power(Decimal('-2'), Decimal('3'))
        Decimal('-8')
        >>> c.power(Decimal('2'), Decimal('-3'))
        Decimal('0.125')
        >>> c.power(Decimal('1.7'), Decimal('8'))
        Decimal('69.7575744')
        >>> c.power(Decimal('10'), Decimal('0.301029996'))
        Decimal('2.00000000')
        >>> c.power(Decimal('Infinity'), Decimal('-1'))
        Decimal('0')
        >>> c.power(Decimal('Infinity'), Decimal('0'))
        Decimal('1')
        >>> c.power(Decimal('Infinity'), Decimal('1'))
        Decimal('Infinity')
        >>> c.power(Decimal('-Infinity'), Decimal('-1'))
        Decimal('-0')
        >>> c.power(Decimal('-Infinity'), Decimal('0'))
        Decimal('1')
        >>> c.power(Decimal('-Infinity'), Decimal('1'))
        Decimal('-Infinity')
        >>> c.power(Decimal('-Infinity'), Decimal('2'))
        Decimal('Infinity')
        >>> c.power(Decimal('0'), Decimal('0'))
        Decimal('NaN')
        
        >>> c.power(Decimal('3'), Decimal('7'), Decimal('16'))
        Decimal('11')
        >>> c.power(Decimal('-3'), Decimal('7'), Decimal('16'))
        Decimal('-11')
        >>> c.power(Decimal('-3'), Decimal('8'), Decimal('16'))
        Decimal('1')
        >>> c.power(Decimal('3'), Decimal('7'), Decimal('-16'))
        Decimal('11')
        >>> c.power(Decimal('23E12345'), Decimal('67E189'), Decimal('123456789'))
        Decimal('11729830')
        >>> c.power(Decimal('-0'), Decimal('17'), Decimal('1729'))
        Decimal('-0')
        >>> c.power(Decimal('-23'), Decimal('0'), Decimal('65537'))
        Decimal('1')
        >>> ExtendedContext.power(7, 7)
        Decimal('823543')
        >>> ExtendedContext.power(Decimal(7), 7)
        Decimal('823543')
        >>> ExtendedContext.power(7, Decimal(7), 2)
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__pow__(b, modulo, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def quantize(self, a, b):
        """Returns a value equal to 'a' (rounded), having the exponent of 'b'.
        
        The coefficient of the result is derived from that of the left-hand
        operand.  It may be rounded using the current rounding setting (if the
        exponent is being increased), multiplied by a positive power of ten (if
        the exponent is being decreased), or is unchanged (if the exponent is
        already equal to that of the right-hand operand).
        
        Unlike other operations, if the length of the coefficient after the
        quantize operation would be greater than precision then an Invalid
        operation condition is raised.  This guarantees that, unless there is
        an error condition, the exponent of the result of a quantize is always
        equal to that of the right-hand operand.
        
        Also unlike other operations, quantize will never raise Underflow, even
        if the result is subnormal and inexact.
        
        >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.001'))
        Decimal('2.170')
        >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.01'))
        Decimal('2.17')
        >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.1'))
        Decimal('2.2')
        >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+0'))
        Decimal('2')
        >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+1'))
        Decimal('0E+1')
        >>> ExtendedContext.quantize(Decimal('-Inf'), Decimal('Infinity'))
        Decimal('-Infinity')
        >>> ExtendedContext.quantize(Decimal('2'), Decimal('Infinity'))
        Decimal('NaN')
        >>> ExtendedContext.quantize(Decimal('-0.1'), Decimal('1'))
        Decimal('-0')
        >>> ExtendedContext.quantize(Decimal('-0'), Decimal('1e+5'))
        Decimal('-0E+5')
        >>> ExtendedContext.quantize(Decimal('+35236450.6'), Decimal('1e-2'))
        Decimal('NaN')
        >>> ExtendedContext.quantize(Decimal('-35236450.6'), Decimal('1e-2'))
        Decimal('NaN')
        >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-1'))
        Decimal('217.0')
        >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-0'))
        Decimal('217')
        >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+1'))
        Decimal('2.2E+2')
        >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+2'))
        Decimal('2E+2')
        >>> ExtendedContext.quantize(1, 2)
        Decimal('1')
        >>> ExtendedContext.quantize(Decimal(1), 2)
        Decimal('1')
        >>> ExtendedContext.quantize(1, Decimal(2))
        Decimal('1')
        """
        a = _convert_other(a, raiseit=True)
        return a.quantize(b, context=self)

    def radix(self):
        """Just returns 10, as this is Decimal, :)
        
        >>> ExtendedContext.radix()
        Decimal('10')
        """
        return Decimal(10)

    def remainder(self, a, b):
        """Returns the remainder from integer division.
        
        The result is the residue of the dividend after the operation of
        calculating integer division as described for divide-integer, rounded
        to precision digits if necessary.  The sign of the result, if
        non-zero, is the same as that of the original dividend.
        
        This operation will fail under the same conditions as integer division
        (that is, if integer division on the same two operands would fail, the
        remainder cannot be calculated).
        
        >>> ExtendedContext.remainder(Decimal('2.1'), Decimal('3'))
        Decimal('2.1')
        >>> ExtendedContext.remainder(Decimal('10'), Decimal('3'))
        Decimal('1')
        >>> ExtendedContext.remainder(Decimal('-10'), Decimal('3'))
        Decimal('-1')
        >>> ExtendedContext.remainder(Decimal('10.2'), Decimal('1'))
        Decimal('0.2')
        >>> ExtendedContext.remainder(Decimal('10'), Decimal('0.3'))
        Decimal('0.1')
        >>> ExtendedContext.remainder(Decimal('3.6'), Decimal('1.3'))
        Decimal('1.0')
        >>> ExtendedContext.remainder(22, 6)
        Decimal('4')
        >>> ExtendedContext.remainder(Decimal(22), 6)
        Decimal('4')
        >>> ExtendedContext.remainder(22, Decimal(6))
        Decimal('4')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__mod__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def remainder_near(self, a, b):
        """Returns to be "a - b * n", where n is the integer nearest the exact
        value of "x / b" (if two integers are equally near then the even one
        is chosen).  If the result is equal to 0 then its sign will be the
        sign of a.
        
        This operation will fail under the same conditions as integer division
        (that is, if integer division on the same two operands would fail, the
        remainder cannot be calculated).
        
        >>> ExtendedContext.remainder_near(Decimal('2.1'), Decimal('3'))
        Decimal('-0.9')
        >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('6'))
        Decimal('-2')
        >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('3'))
        Decimal('1')
        >>> ExtendedContext.remainder_near(Decimal('-10'), Decimal('3'))
        Decimal('-1')
        >>> ExtendedContext.remainder_near(Decimal('10.2'), Decimal('1'))
        Decimal('0.2')
        >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('0.3'))
        Decimal('0.1')
        >>> ExtendedContext.remainder_near(Decimal('3.6'), Decimal('1.3'))
        Decimal('-0.3')
        >>> ExtendedContext.remainder_near(3, 11)
        Decimal('3')
        >>> ExtendedContext.remainder_near(Decimal(3), 11)
        Decimal('3')
        >>> ExtendedContext.remainder_near(3, Decimal(11))
        Decimal('3')
        """
        a = _convert_other(a, raiseit=True)
        return a.remainder_near(b, context=self)

    def rotate(self, a, b):
        """Returns a rotated copy of a, b times.
        
        The coefficient of the result is a rotated copy of the digits in
        the coefficient of the first operand.  The number of places of
        rotation is taken from the absolute value of the second operand,
        with the rotation being to the left if the second operand is
        positive or to the right otherwise.
        
        >>> ExtendedContext.rotate(Decimal('34'), Decimal('8'))
        Decimal('400000003')
        >>> ExtendedContext.rotate(Decimal('12'), Decimal('9'))
        Decimal('12')
        >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('-2'))
        Decimal('891234567')
        >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('0'))
        Decimal('123456789')
        >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('+2'))
        Decimal('345678912')
        >>> ExtendedContext.rotate(1333333, 1)
        Decimal('13333330')
        >>> ExtendedContext.rotate(Decimal(1333333), 1)
        Decimal('13333330')
        >>> ExtendedContext.rotate(1333333, Decimal(1))
        Decimal('13333330')
        """
        a = _convert_other(a, raiseit=True)
        return a.rotate(b, context=self)

    def same_quantum(self, a, b):
        """Returns True if the two operands have the same exponent.
        
        The result is never affected by either the sign or the coefficient of
        either operand.
        
        >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.001'))
        False
        >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.01'))
        True
        >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('1'))
        False
        >>> ExtendedContext.same_quantum(Decimal('Inf'), Decimal('-Inf'))
        True
        >>> ExtendedContext.same_quantum(10000, -1)
        True
        >>> ExtendedContext.same_quantum(Decimal(10000), -1)
        True
        >>> ExtendedContext.same_quantum(10000, Decimal(-1))
        True
        """
        a = _convert_other(a, raiseit=True)
        return a.same_quantum(b)

    def scaleb(self, a, b):
        """Returns the first operand after adding the second value its exp.
        
        >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('-2'))
        Decimal('0.0750')
        >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('0'))
        Decimal('7.50')
        >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('3'))
        Decimal('7.50E+3')
        >>> ExtendedContext.scaleb(1, 4)
        Decimal('1E+4')
        >>> ExtendedContext.scaleb(Decimal(1), 4)
        Decimal('1E+4')
        >>> ExtendedContext.scaleb(1, Decimal(4))
        Decimal('1E+4')
        """
        a = _convert_other(a, raiseit=True)
        return a.scaleb(b, context=self)

    def shift(self, a, b):
        """Returns a shifted copy of a, b times.
        
        The coefficient of the result is a shifted copy of the digits
        in the coefficient of the first operand.  The number of places
        to shift is taken from the absolute value of the second operand,
        with the shift being to the left if the second operand is
        positive or to the right otherwise.  Digits shifted into the
        coefficient are zeros.
        
        >>> ExtendedContext.shift(Decimal('34'), Decimal('8'))
        Decimal('400000000')
        >>> ExtendedContext.shift(Decimal('12'), Decimal('9'))
        Decimal('0')
        >>> ExtendedContext.shift(Decimal('123456789'), Decimal('-2'))
        Decimal('1234567')
        >>> ExtendedContext.shift(Decimal('123456789'), Decimal('0'))
        Decimal('123456789')
        >>> ExtendedContext.shift(Decimal('123456789'), Decimal('+2'))
        Decimal('345678900')
        >>> ExtendedContext.shift(88888888, 2)
        Decimal('888888800')
        >>> ExtendedContext.shift(Decimal(88888888), 2)
        Decimal('888888800')
        >>> ExtendedContext.shift(88888888, Decimal(2))
        Decimal('888888800')
        """
        a = _convert_other(a, raiseit=True)
        return a.shift(b, context=self)

    def sqrt(self, a):
        """Square root of a non-negative number to context precision.
        
        If the result must be inexact, it is rounded using the round-half-even
        algorithm.
        
        >>> ExtendedContext.sqrt(Decimal('0'))
        Decimal('0')
        >>> ExtendedContext.sqrt(Decimal('-0'))
        Decimal('-0')
        >>> ExtendedContext.sqrt(Decimal('0.39'))
        Decimal('0.624499800')
        >>> ExtendedContext.sqrt(Decimal('100'))
        Decimal('10')
        >>> ExtendedContext.sqrt(Decimal('1'))
        Decimal('1')
        >>> ExtendedContext.sqrt(Decimal('1.0'))
        Decimal('1.0')
        >>> ExtendedContext.sqrt(Decimal('1.00'))
        Decimal('1.0')
        >>> ExtendedContext.sqrt(Decimal('7'))
        Decimal('2.64575131')
        >>> ExtendedContext.sqrt(Decimal('10'))
        Decimal('3.16227766')
        >>> ExtendedContext.sqrt(2)
        Decimal('1.41421356')
        >>> ExtendedContext.prec
        9
        """
        a = _convert_other(a, raiseit=True)
        return a.sqrt(context=self)

    def subtract(self, a, b):
        """Return the difference between the two operands.
        
        >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.07'))
        Decimal('0.23')
        >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.30'))
        Decimal('0.00')
        >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('2.07'))
        Decimal('-0.77')
        >>> ExtendedContext.subtract(8, 5)
        Decimal('3')
        >>> ExtendedContext.subtract(Decimal(8), 5)
        Decimal('3')
        >>> ExtendedContext.subtract(8, Decimal(5))
        Decimal('3')
        """
        a = _convert_other(a, raiseit=True)
        r = a.__sub__(b, context=self)
        if r is NotImplemented:
            raise TypeError("Unable to convert %s to Decimal" % b)
        else:
            return r

    def to_eng_string(self, a):
        """Convert to a string, using engineering notation if an exponent is needed.
        
        Engineering notation has an exponent which is a multiple of 3.  This
        can leave up to 3 digits to the left of the decimal place and may
        require the addition of either one or two trailing zeros.
        
        The operation is not affected by the context.
        
        >>> ExtendedContext.to_eng_string(Decimal('123E+1'))
        '1.23E+3'
        >>> ExtendedContext.to_eng_string(Decimal('123E+3'))
        '123E+3'
        >>> ExtendedContext.to_eng_string(Decimal('123E-10'))
        '12.3E-9'
        >>> ExtendedContext.to_eng_string(Decimal('-123E-12'))
        '-123E-12'
        >>> ExtendedContext.to_eng_string(Decimal('7E-7'))
        '700E-9'
        >>> ExtendedContext.to_eng_string(Decimal('7E+1'))
        '70'
        >>> ExtendedContext.to_eng_string(Decimal('0E+1'))
        '0.00E+3'
        
        """
        a = _convert_other(a, raiseit=True)
        return a.to_eng_string(context=self)

    def to_sci_string(self, a):
        """Converts a number to a string, using scientific notation.
        
        The operation is not affected by the context.
        """
        a = _convert_other(a, raiseit=True)
        return a.__str__(context=self)

    def to_integral_exact(self, a):
        """Rounds to an integer.
        
        When the operand has a negative exponent, the result is the same
        as using the quantize() operation using the given operand as the
        left-hand-operand, 1E+0 as the right-hand-operand, and the precision
        of the operand as the precision setting; Inexact and Rounded flags
        are allowed in this operation.  The rounding mode is taken from the
        context.
        
        >>> ExtendedContext.to_integral_exact(Decimal('2.1'))
        Decimal('2')
        >>> ExtendedContext.to_integral_exact(Decimal('100'))
        Decimal('100')
        >>> ExtendedContext.to_integral_exact(Decimal('100.0'))
        Decimal('100')
        >>> ExtendedContext.to_integral_exact(Decimal('101.5'))
        Decimal('102')
        >>> ExtendedContext.to_integral_exact(Decimal('-101.5'))
        Decimal('-102')
        >>> ExtendedContext.to_integral_exact(Decimal('10E+5'))
        Decimal('1.0E+6')
        >>> ExtendedContext.to_integral_exact(Decimal('7.89E+77'))
        Decimal('7.89E+77')
        >>> ExtendedContext.to_integral_exact(Decimal('-Inf'))
        Decimal('-Infinity')
        """
        a = _convert_other(a, raiseit=True)
        return a.to_integral_exact(context=self)

    def to_integral_value(self, a):
        """Rounds to an integer.
        
        When the operand has a negative exponent, the result is the same
        as using the quantize() operation using the given operand as the
        left-hand-operand, 1E+0 as the right-hand-operand, and the precision
        of the operand as the precision setting, except that no flags will
        be set.  The rounding mode is taken from the context.
        
        >>> ExtendedContext.to_integral_value(Decimal('2.1'))
        Decimal('2')
        >>> ExtendedContext.to_integral_value(Decimal('100'))
        Decimal('100')
        >>> ExtendedContext.to_integral_value(Decimal('100.0'))
        Decimal('100')
        >>> ExtendedContext.to_integral_value(Decimal('101.5'))
        Decimal('102')
        >>> ExtendedContext.to_integral_value(Decimal('-101.5'))
        Decimal('-102')
        >>> ExtendedContext.to_integral_value(Decimal('10E+5'))
        Decimal('1.0E+6')
        >>> ExtendedContext.to_integral_value(Decimal('7.89E+77'))
        Decimal('7.89E+77')
        >>> ExtendedContext.to_integral_value(Decimal('-Inf'))
        Decimal('-Infinity')
        """
        a = _convert_other(a, raiseit=True)
        return a.to_integral_value(context=self)

    to_integral = to_integral_value


class _WorkRep(object):
    __slots__ = ("sign", "int", "exp")

    def __init__(self, value=None):
        if value is None:
            self.sign = None
            self.int = 0
            self.exp = None
        else:
            if isinstance(value, Decimal):
                self.sign = value._sign
                self.int = int(value._int)
                self.exp = value._exp
            else:
                self.sign = value[0]
                self.int = value[1]
                self.exp = value[2]

    def __repr__(self):
        return "(%r, %r, %r)" % (self.sign, self.int, self.exp)

    __str__ = __repr__


def _normalize(op1, op2, prec=0):
    """Normalizes op1, op2 to have the same exp and length of coefficient.
    
    Done during addition.
    """
    if op1.exp < op2.exp:
        tmp = op2
        other = op1
    else:
        tmp = op1
        other = op2
    tmp_len = len(str(tmp.int))
    other_len = len(str(other.int))
    exp = tmp.exp + min(-1, tmp_len - prec - 2)
    if other_len + other.exp - 1 < exp:
        other.int = 1
        other.exp = exp
    tmp.int *= 10 ** (tmp.exp - other.exp)
    tmp.exp = other.exp
    return (op1, op2)


_nbits = int.bit_length


def _decimal_lshift_exact(n, e):
    """ Given integers n and e, return n * 10**e if it's an integer, else None.
    
    The computation is designed to avoid computing large powers of 10
    unnecessarily.
    
    >>> _decimal_lshift_exact(3, 4)
    30000
    >>> _decimal_lshift_exact(300, -999999999)  # returns None
    
    """
    if n == 0:
        return 0
    elif e >= 0:
        return n * 10 ** e
    str_n = str(abs(n))
    val_n = len(str_n) - len(str_n.rstrip("0"))
    if val_n < -e:
        return
    else:
        return n // 10 ** (-e)


def _sqrt_nearest(n, a):
    """Closest integer to the square root of the positive integer n.  a is
    an initial approximation to the square root.  Any positive integer
    will do for a, but the closer a is to the square root of n the
    faster convergence will be.
    
    """
    if n <= 0 or a <= 0:
        raise ValueError("Both arguments to _sqrt_nearest should be positive.")
    b = 0
    while a != b:
        b, a = a, a - -n // a >> 1

    return a


def _rshift_nearest(x, shift):
    """Given an integer x and a nonnegative integer shift, return closest
    integer to x / 2**shift; use round-to-even in case of a tie.
    
    """
    b, q = 1 << shift, x >> shift
    return q + (2 * (x & b - 1) + (q & 1) > b)


def _div_nearest(a, b):
    """Closest integer to a/b, a and b positive integers; rounds to even
    in the case of a tie.
    
    """
    q, r = divmod(a, b)
    return q + (2 * r + (q & 1) > b)


def _ilog(x, M, L=8):
    """Integer approximation to M*log(x/M), with absolute error boundable
    in terms only of x/M.
    
    Given positive integers x and M, return an integer approximation to
    M * log(x/M).  For L = 8 and 0.1 <= x/M <= 10 the difference
    between the approximation and the exact result is at most 22.  For
    L = 8 and 1.0 <= x/M <= 10.0 the difference is at most 15.  In
    both cases these are upper bounds on the error; it will usually be
    much smaller."""
    y = x - M
    R = 0
    while R <= L and abs(y) << L - R >= M or R > L and abs(y) >> R - L >= M:
        y = _div_nearest(
            M * y << 1, M + _sqrt_nearest(M * (M + _rshift_nearest(y, R)), M)
        )
        R += 1

    T = -int(-10 * len(str(M)) // (3 * L))
    yshift = _rshift_nearest(y, R)
    w = _div_nearest(M, T)
    for k in range(T - 1, 0, -1):
        w = _div_nearest(M, k) - _div_nearest(yshift * w, M)

    return _div_nearest(w * y, M)


def _dlog10(c, e, p):
    """Given integers c, e and p with c > 0, p >= 0, compute an integer
    approximation to 10**p * log10(c*10**e), with an absolute error of
    at most 1.  Assumes that c*10**e is not exactly 1."""
    p += 2
    l = len(str(c))
    f = e + l - (e + l >= 1)
    if p > 0:
        M = 10 ** p
        k = e + p - f
        if k >= 0:
            c *= 10 ** k
        else:
            c = _div_nearest(c, 10 ** (-k))
        log_d = _ilog(c, M)
        log_10 = _log10_digits(p)
        log_d = _div_nearest(log_d * M, log_10)
        log_tenpower = f * M
    else:
        log_d = 0
        log_tenpower = _div_nearest(f, 10 ** (-p))
    return _div_nearest(log_tenpower + log_d, 100)


def _dlog(c, e, p):
    """Given integers c, e and p with c > 0, compute an integer
    approximation to 10**p * log(c*10**e), with an absolute error of
    at most 1.  Assumes that c*10**e is not exactly 1."""
    p += 2
    l = len(str(c))
    f = e + l - (e + l >= 1)
    if p > 0:
        k = e + p - f
        if k >= 0:
            c *= 10 ** k
        else:
            c = _div_nearest(c, 10 ** (-k))
        log_d = _ilog(c, 10 ** p)
    else:
        log_d = 0
    if f:
        extra = len(str(abs(f))) - 1
        if p + extra >= 0:
            f_log_ten = _div_nearest(f * _log10_digits(p + extra), 10 ** extra)
        else:
            f_log_ten = 0
    else:
        f_log_ten = 0
    return _div_nearest(f_log_ten + log_d, 100)


class _Log10Memoize(object):
    """Class to compute, store, and allow retrieval of, digits of the
    constant log(10) = 2.302585....  This constant is needed by
    Decimal.ln, Decimal.log10, Decimal.exp and Decimal.__pow__."""

    def __init__(self):
        self.digits = "23025850929940456840179914546843642076011014886"

    def getdigits(self, p):
        """Given an integer p >= 0, return floor(10**p)*log(10).
        
        For example, self.getdigits(3) returns 2302.
        """
        if p < 0:
            raise ValueError("p should be nonnegative")
        if p >= len(self.digits):
            extra = 3
            while True:
                M = 10 ** (p + extra + 2)
                digits = str(_div_nearest(_ilog(10 * M, M), 100))
                if digits[-extra:] != "0" * extra:
                    break
                extra += 3

            self.digits = digits.rstrip("0")[:-1]
        return int(self.digits[: p + 1])


_log10_digits = _Log10Memoize().getdigits


def _iexp(x, M, L=8):
    """Given integers x and M, M > 0, such that x/M is small in absolute
    value, compute an integer approximation to M*exp(x/M).  For 0 <=
    x/M <= 2.4, the absolute error in the result is bounded by 60 (and
    is usually much smaller)."""
    R = _nbits((x << L) // M)
    T = -int(-10 * len(str(M)) // (3 * L))
    y = _div_nearest(x, T)
    Mshift = M << R
    for i in range(T - 1, 0, -1):
        y = _div_nearest(x * (Mshift + y), Mshift * i)

    for k in range(R - 1, -1, -1):
        Mshift = M << k + 2
        y = _div_nearest(y * (y + Mshift), Mshift)

    return M + y


def _dexp(c, e, p):
    """Compute an approximation to exp(c*10**e), with p decimal places of
    precision.
    
    Returns integers d, f such that:
    
      10**(p-1) <= d <= 10**p, and
      (d-1)*10**f < exp(c*10**e) < (d+1)*10**f
    
    In other words, d*10**f is an approximation to exp(c*10**e) with p
    digits of precision, and with an error in d of at most 1.  This is
    almost, but not quite, the same as the error being < 1ulp: when d
    = 10**(p-1) the error could be up to 10 ulp."""
    p += 2
    extra = max(0, e + len(str(c)) - 1)
    q = p + extra
    shift = e + q
    if shift >= 0:
        cshift = c * 10 ** shift
    else:
        cshift = c // 10 ** (-shift)
    quot, rem = divmod(cshift, _log10_digits(q))
    rem = _div_nearest(rem, 10 ** extra)
    return (_div_nearest(_iexp(rem, 10 ** p), 1000), quot - p + 3)


def _dpower(xc, xe, yc, ye, p):
    """Given integers xc, xe, yc and ye representing Decimals x = xc*10**xe and
    y = yc*10**ye, compute x**y.  Returns a pair of integers (c, e) such that:
    
      10**(p-1) <= c <= 10**p, and
      (c-1)*10**e < x**y < (c+1)*10**e
    
    in other words, c*10**e is an approximation to x**y with p digits
    of precision, and with an error in c of at most 1.  (This is
    almost, but not quite, the same as the error being < 1ulp: when c
    == 10**(p-1) we can only guarantee error < 10ulp.)
    
    We assume that: x is positive and not equal to 1, and y is nonzero.
    """
    b = len(str(abs(yc))) + ye
    lxc = _dlog(xc, xe, p + b + 1)
    shift = ye - b
    if shift >= 0:
        pc = lxc * yc * 10 ** shift
    else:
        pc = _div_nearest(lxc * yc, 10 ** (-shift))
    if pc == 0:
        if (len(str(xc)) + xe >= 1) == (yc > 0):
            coeff, exp = 10 ** (p - 1) + 1, 1 - p
        else:
            coeff, exp = 10 ** p - 1, -p
    else:
        coeff, exp = _dexp(pc, -(p + 1), p + 1)
        coeff = _div_nearest(coeff, 10)
        exp += 1
    return (coeff, exp)


def _log10_lb(
    c,
    correction={
        "1": 100,
        "2": 70,
        "3": 53,
        "4": 40,
        "5": 31,
        "6": 23,
        "7": 16,
        "8": 10,
        "9": 5,
    },
):
    """Compute a lower bound for 100*log10(c) for a positive integer c."""
    if c <= 0:
        raise ValueError("The argument to _log10_lb should be nonnegative.")
    str_c = str(c)
    return 100 * len(str_c) - correction[str_c[0]]


def _convert_other(other, raiseit=False, allow_float=False):
    """Convert other to Decimal.
    
    Verifies that it's ok to use in an implicit construction.
    If allow_float is true, allow conversion from float;  this
    is used in the comparison methods (__eq__ and friends).
    
    """
    if isinstance(other, Decimal):
        return other
    elif isinstance(other, int):
        return Decimal(other)
    else:
        if allow_float:
            if isinstance(other, float):
                return Decimal.from_float(other)
            if raiseit:
                raise TypeError("Unable to convert %s to Decimal" % other)
        return NotImplemented


def _convert_for_comparison(self, other, equality_op=False):
    """Given a Decimal instance self and a Python object other, return
    a pair (s, o) of Decimal instances such that "s op o" is
    equivalent to "self op other" for any of the 6 comparison
    operators "op".
    
    """
    if isinstance(other, Decimal):
        return (self, other)
    elif isinstance(other, _numbers.Rational):
        if not self._is_special:
            self = _dec_from_triple(
                self._sign, str(int(self._int) * other.denominator), self._exp
            )
        return (self, Decimal(other.numerator))
    if equality_op:
        if isinstance(other, _numbers.Complex):
            if other.imag == 0:
                other = other.real
    if isinstance(other, float):
        context = getcontext()
        if equality_op:
            context.flags[FloatOperation] = 1
        else:
            context._raise_error(
                FloatOperation,
                "strict semantics for mixing floats and Decimals are enabled",
            )
        return (self, Decimal.from_float(other))
    else:
        return (NotImplemented, NotImplemented)


DefaultContext = Context(
    prec=28,
    rounding=ROUND_HALF_EVEN,
    traps=[DivisionByZero, Overflow, InvalidOperation],
    flags=[],
    Emax=999999,
    Emin=-999999,
    capitals=1,
    clamp=0,
)
BasicContext = Context(
    prec=9,
    rounding=ROUND_HALF_UP,
    traps=[DivisionByZero, Overflow, InvalidOperation, Clamped, Underflow],
    flags=[],
)
ExtendedContext = Context(prec=9, rounding=ROUND_HALF_EVEN, traps=[], flags=[])
import re

_parser = re.compile(
    "        # A numeric string consists of:\n#    \\s*\n    (?P<sign>[-+])?              # an optional sign, followed by either...\n    (\n        (?=\\d|\\.\\d)              # ...a number (with at least one digit)\n        (?P<int>\\d*)             # having a (possibly empty) integer part\n        (\\.(?P<frac>\\d*))?       # followed by an optional fractional part\n        (E(?P<exp>[-+]?\\d+))?    # followed by an optional exponent, or...\n    |\n        Inf(inity)?              # ...an infinity, or...\n    |\n        (?P<signal>s)?           # ...an (optionally signaling)\n        NaN                      # NaN\n        (?P<diag>\\d*)            # with (possibly empty) diagnostic info.\n    )\n#    \\s*\n    \\Z\n",
    re.VERBOSE | re.IGNORECASE,
).match
_all_zeros = re.compile("0*$").match
_exact_half = re.compile("50*$").match
_parse_format_specifier_regex = re.compile(
    "\\A\n(?:\n   (?P<fill>.)?\n   (?P<align>[<>=^])\n)?\n(?P<sign>[-+ ])?\n(?P<alt>\\#)?\n(?P<zeropad>0)?\n(?P<minimumwidth>(?!0)\\d+)?\n(?P<thousands_sep>,)?\n(?:\\.(?P<precision>0|(?!0)\\d+))?\n(?P<type>[eEfFgGn%])?\n\\Z\n",
    re.VERBOSE | re.DOTALL,
)
del re
try:
    import locale as _locale
except ImportError:
    pass


def _parse_format_specifier(format_spec, _localeconv=None):
    """Parse and validate a format specifier.
    
    Turns a standard numeric format specifier into a dict, with the
    following entries:
    
      fill: fill character to pad field to minimum width
      align: alignment type, either '<', '>', '=' or '^'
      sign: either '+', '-' or ' '
      minimumwidth: nonnegative integer giving minimum width
      zeropad: boolean, indicating whether to pad with zeros
      thousands_sep: string to use as thousands separator, or ''
      grouping: grouping for thousands separators, in format
        used by localeconv
      decimal_point: string to use for decimal point
      precision: nonnegative integer giving precision, or None
      type: one of the characters 'eEfFgG%', or None
    
    """
    m = _parse_format_specifier_regex.match(format_spec)
    if m is None:
        raise ValueError("Invalid format specifier: " + format_spec)
    format_dict = m.groupdict()
    fill = format_dict["fill"]
    align = format_dict["align"]
    format_dict["zeropad"] = format_dict["zeropad"] is not None
    if format_dict["zeropad"]:
        if fill is not None:
            raise ValueError(
                "Fill character conflicts with '0' in format specifier: " + format_spec
            )
        if align is not None:
            raise ValueError(
                "Alignment conflicts with '0' in format specifier: " + format_spec
            )
    format_dict["fill"] = fill or " "
    format_dict["align"] = align or ">"
    if format_dict["sign"] is None:
        format_dict["sign"] = "-"
    format_dict["minimumwidth"] = int(format_dict["minimumwidth"] or "0")
    if format_dict["precision"] is not None:
        format_dict["precision"] = int(format_dict["precision"])
    if format_dict["precision"] == 0:
        if format_dict["type"] is None or format_dict["type"] in "gGn":
            format_dict["precision"] = 1
    if format_dict["type"] == "n":
        format_dict["type"] = "g"
        if _localeconv is None:
            _localeconv = _locale.localeconv()
        if format_dict["thousands_sep"] is not None:
            raise ValueError(
                "Explicit thousands separator conflicts with 'n' type in format specifier: "
                + format_spec
            )
        format_dict["thousands_sep"] = _localeconv["thousands_sep"]
        format_dict["grouping"] = _localeconv["grouping"]
        format_dict["decimal_point"] = _localeconv["decimal_point"]
    else:
        if format_dict["thousands_sep"] is None:
            format_dict["thousands_sep"] = ""
        format_dict["grouping"] = [3, 0]
        format_dict["decimal_point"] = "."
    return format_dict


def _format_align(sign, body, spec):
    """Given an unpadded, non-aligned numeric string 'body' and sign
    string 'sign', add padding and alignment conforming to the given
    format specifier dictionary 'spec' (as produced by
    parse_format_specifier).
    
    """
    minimumwidth = spec["minimumwidth"]
    fill = spec["fill"]
    padding = fill * (minimumwidth - len(sign) - len(body))
    align = spec["align"]
    if align == "<":
        result = sign + body + padding
    else:
        if align == ">":
            result = padding + sign + body
        else:
            if align == "=":
                result = sign + padding + body
            else:
                if align == "^":
                    half = len(padding) // 2
                    result = padding[:half] + sign + body + padding[half:]
                else:
                    raise ValueError("Unrecognised alignment field")
                return result


def _group_lengths(grouping):
    """Convert a localeconv-style grouping into a (possibly infinite)
    iterable of integers representing group lengths.
    
    """
    from itertools import chain, repeat

    if not grouping:
        return []
    if grouping[-1] == 0:
        if len(grouping) >= 2:
            return chain(grouping[:-1], repeat(grouping[-2]))
        if grouping[-1] == _locale.CHAR_MAX:
            return grouping[:-1]
        raise ValueError("unrecognised format for grouping")


def _insert_thousands_sep(digits, spec, min_width=1):
    """Insert thousands separators into a digit string.
    
    spec is a dictionary whose keys should include 'thousands_sep' and
    'grouping'; typically it's the result of parsing the format
    specifier using _parse_format_specifier.
    
    The min_width keyword argument gives the minimum length of the
    result, which will be padded on the left with zeros if necessary.
    
    If necessary, the zero padding adds an extra '0' on the left to
    avoid a leading thousands separator.  For example, inserting
    commas every three digits in '123456', with min_width=8, gives
    '0,123,456', even though that has length 9.
    
    """
    sep = spec["thousands_sep"]
    grouping = spec["grouping"]
    groups = []
    for l in _group_lengths(grouping):
        if l <= 0:
            raise ValueError("group length should be positive")
        l = min(max(len(digits), min_width, 1), l)
        groups.append("0" * (l - len(digits)) + digits[-l:])
        digits = digits[:-l]
        min_width -= l
        if not digits:
            if min_width <= 0:
                break
        min_width -= len(sep)
    else:
        l = max(len(digits), min_width, 1)
        groups.append("0" * (l - len(digits)) + digits[-l:])

    return sep.join(reversed(groups))


def _format_sign(is_negative, spec):
    """Determine sign character."""
    if is_negative:
        return "-"
    elif spec["sign"] in " +":
        return spec["sign"]
    else:
        return ""


def _format_number(is_negative, intpart, fracpart, exp, spec):
    """Format a number, given the following data:
    
    is_negative: true if the number is negative, else false
    intpart: string of digits that must appear before the decimal point
    fracpart: string of digits that must come after the point
    exp: exponent, as an integer
    spec: dictionary resulting from parsing the format specifier
    
    This function uses the information in spec to:
      insert separators (decimal separator and thousands separators)
      format the sign
      format the exponent
      add trailing '%' for the '%' type
      zero-pad if necessary
      fill and align if necessary
    """
    sign = _format_sign(is_negative, spec)
    if fracpart or spec["alt"]:
        fracpart = spec["decimal_point"] + fracpart
    if exp != 0 or spec["type"] in "eE":
        echar = {"E": "E", "e": "e", "G": "E", "g": "e"}[spec["type"]]
        fracpart += ("{0}{1:+}").format(echar, exp)
    if spec["type"] == "%":
        fracpart += "%"
    if spec["zeropad"]:
        min_width = spec["minimumwidth"] - len(fracpart) - len(sign)
    else:
        min_width = 0
    intpart = _insert_thousands_sep(intpart, spec, min_width)
    return _format_align(sign, intpart + fracpart, spec)


_Infinity = Decimal("Inf")
_NegativeInfinity = Decimal("-Inf")
_NaN = Decimal("NaN")
_Zero = Decimal(0)
_One = Decimal(1)
_NegativeOne = Decimal(-1)
_SignedInfinity = (_Infinity, _NegativeInfinity)
_PyHASH_MODULUS = sys.hash_info.modulus
_PyHASH_INF = sys.hash_info.inf
_PyHASH_NAN = sys.hash_info.nan
_PyHASH_10INV = pow(10, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
del sys
